Wilson outside the pawn shop from ten to two, daily, at the cost of four sovereigns per week. The next question then is why Spaulding wants his employer away from the shop at such hours so much so that he is even willing to part with four sovereigns every week. Since Mr. Wilson’s business is small and barely profitable, Holmes thinks that the attraction must be something outside of the business. Also, the fact that Mr. Wilson’s description of Spaulding answers to that of John Clay, a “murderer, thief, smasher, and forger” (whom, however, Holmes has never met), makes Spaulding a prime suspect in the case. Following Mr. Wilson’s earlier admission that Spaulding spent hours in the shop’s cellar, together with evidence of Spaulding’s worn and dirty trouser knees, Holmes concludes that the assistant spends his time digging at something in the cellar. The only plausible explanation then is that Spaulding must be digging a tunnel in Wilson’s cellar, a tunnel he hopes to use to get to another business situated close to the pawn shop, the most lucrative hit then being the City and Suburban Bank.

Holmes also observed:

1. The Red-Headed League appeared shortly after Jabez hired Spalding. 
2. Spalding encouraged Jabez to remain in line when he found everyone in the city with a trace of red in their hair had responded to the ad in the paper.

These clues told Holmes that Spalding may not be who he seemed to be.  The clues listed by crmhaske identified confirmed Spalding's identity.  The 5 clues related to what Spalding was doing, and the shop's location relative to the bank told Holmes what Spalding and the fictitious Red-Headed League were up to. 

The three clues that lead Holmes to conclude who Spalding's real identity is are:

1. Appearance: short, stout, quick, and no facial hair
2. There is a white splash of acid on his forehead
3. Pierced ears for wearing earings from a Gypsy

There are five clues in total in the novel that Holmes uses to solve the mystery:

1. Spalding was frequently in the cellar
2. From 10am-2pm, Jabez was at his job with the red-headed league
3. The ground in front of Jabez's pawn shop is solid
4. Spalding's knees were covered with soil
5. Jabez's pawn shop is located behind the City and Suburban Bank

This question is easy to answer because Sherlock Holmes explains his whole line of thinking to his friend Dr. Watson at the end of the story, after the burglars have been arrested and the two men are back at Baker Street.

"You see, Watson," he explained in the early hours of the morning as we sat over a glass of whisky and soda in Baker Street, "it was perfectly obvious from the first that the only possible object of this rather fantastic business of the advertisement of the League, and the copying of the 'Encyclopaedia,' must be to get this not over-bright pawnbroker out of the way for a number of hours every day.

That was the first clue. Holmes goes on to explain that Vincent Spaulding's (John Clay's) willingness to work for half wages showed that he had a strong motive for securing that particular situation. His trick of vanishing into the cellar was another clue.

"He was doing something in the cellar--something which took many hours a day for months on end. . . . I could think of nothing save that he was running a tunnel to some other building."

The fourth clue was the condition of Clay's trousers when Holmes rang the shop bell on the pretense of inquiring how to get to the Strand. The knees were "worn, wrinkled, and stained."

The fifth important clue was that they closed the League offices, showing that they no longer cared about getting Jabez Wilson out of the way and that they were probably going to make their burglary attempt on the bank that weekend.

Two other clues which Holmes does not mention in his summation occur earlier in the story at the end of the initial interview with Jabez Wilson. Holmes asks Wilson:

"What is he like, this Vincent Spaulding?"

After Wilson finishes describing his assistant, including that he has a "white splash of acid upon his forehead," Holmes asks:

"Have you ever observed that his ears are pierced for earrings?"

They are. From Wilson's description, Holmes obviously recognizes the real identity of this "Vincent Spaulding." Wilson's assistant has to be John Clay, a notorious criminal with whom Holmes has had dealings in the past. Holmes, however, does not share his thoughts with either Watson or Wilson--or with the reader. Spaulding's true identity does not come out until Holmes tells the bank director, Mr. Merryweather, that they are setting a trap for:

"John Clay, the murderer, thief, smasher, and forger. . . . I've been on his track for years and have never set eyes on him yet."

Then after a long wait in the dark, John Clay is apprehended as he climbs out of his tunnel.

"It's no use, John Clay," said Holmes blandly. "You have no chance at all."

Clay will be hanged for the attempted crime, as we know when he shouts:

"Great Scott! Jump, Archie, jump, and I'll swing for it!"

Many criminal offenses were subject to capital punishment in Victorian times, as we know from reading Charles Dickens' novels, especially Oliver Twist and Great Expectations. But Clay would have other crimes to pay for now that he is in the hands of the law.

The first thing that arouses Holmes's suspicions is in hearing that Mr Jabez Wilson's new assistant in his shop, Vincent Spaulding, was perfectly willing to work for half the usual wage. This sets Holmes's focus upon him from the beginning. It is also Spaulding who first attracted Wilson's attention to the whole extraordinary business of the Red-Headed League. Furthermore, on learning that Spaulding habitually spends most of his waking time in the basement, claiming that he uses it as a darkroom for his photography, Holmes wonders what his real motive is in spending so much time down there. Therefore, Holmes makes a point of setting himself upon Spaulding's track and discovers that he really is John Clay, a known and crafty criminal, and that he has been secretly tunnelling his way from Wilson's basement to the bank. Spaulding is the key to the whole case in this story, and once Holmes fixes on him, it does not take him long to solve the whole mystery.

Secondly, the utterly bizarre nature of the Red-Headed League itself arouses suspicions. On learning that Wilson normally doesn't venture outside, Holmes realises that the whole strange affair has been devised by Clay and his confederates simply to get Wilson out of his shop, on the pretext of easy employment. It is rather an extreme way to go about it, but as Holmes frequently remarks in these stories, the more bizarre cases are generally easier to solve than commonplace ones. 

This story is one of the more light-hearted ones in the Holmes canon. Jabez Wilson, with his shock flaming red hair and peculiar tale, comes across as a rather comic figure; indeed, at one point both Holmes and Watson laugh unceremoniously at his expense. Wilson is vexed at this but Holmes pacifies him by remarking on how much the case appeals:

I really wouldn't miss your case for the world. It is most refreshingly unusual.

Holmes deduced that The Red-Headed League did not exist, and it was actually nothing more than a ruse to get Mr. Wilson out of his shop so Clay and his accomplice could dig a tunnel.

Mr. Wilson came to Sherlock Holmes upset and confused because his employer had closed up shop.  It turned out he had answered an advertisement for men with red hair, supposedly from The Red-Headed League.  He was hired to sit in an office and not leave, doing nothing but copying the encyclopedia.

Holmes took the case, because he knew something strange was going on.  He visited the shop, and realized that it was near a bank.  He knew then from the dirt on the assistant’s knees and the hollow feel of the ground that someone was digging a tunnel.

Watson was really confused, especially when Holmes confirmed his deduction by tapping on the pavement with his stick.

Finally he returned to the pawnbroker's, and, having thumped vigorously upon the pavement with his stick two or three times, he went up to the door and knocked. It was instantly opened by a bright-looking, clean-shaven young fellow, who asked him to step in.

Holmes said that John Clay was a notorious and brilliant criminal.  He arranged for the police to meet him in the bank, having deduced that the tunnel was done and the men would act soon.  The police came, and they caught the robbers red-handed as they tried to break in.

It took not just amazing attention to detail but an intimate knowledge of the London criminal scene to catch Clay.  Holmes knew who he was dealing with at once.  He realized Clay was smart, and deduced the best way to catch him- or at least the most exciting!

In “The Red-Headed League” Holmes uses logic to solve a seemingly bizarre case of a man who is given a job copying the encyclopedia because he realizes that the suspects are trying to get the man out of his shop so they can make a tunnel to a nearby bank.

In this story a pawnbroker named Mr. Wilson is sponsored by the “Red-Headed League” an organization supposedly started by a red-head in America.  "Holmes possesses a nearly superhuman ability to read a person's background by observing small, seemingly-insignificant details" (enotes, characters).

The Strange Advertisement

Holmes’s first clue is the story itself.  The “Red-Headed League” is just too crazy to be real.  Holmes knows that there is some kind of ulterior motive.  Since the advertisement describes an American, there is no real way to check the story.

Holmes sees that the story is not meant to be checked, so it is likely not legitimate.

The Overly-Helpful Assistant

The second clue is the assistant.  Holmes is immediately curious about him, and believes that he is “as remarkable” as the advertisement for the Red-Headed League.  He is the one that brings Mr. Wilson the advertisement about the Red-Headed league and encourages him to apply, and he showed up just before the message.

You seem most fortunate in having an employee who comes under the full market price. (enotes etext pdf, p. 6)

Besides showing up out of nowhere, being too old to be an assistant, and working for too little money, Mr. Wilson also comments on other strange facts about his assistant.  He has a habit of taking pictures and then “diving down into the cellar like a rabbit into its hole to develop his pictures” (p. 6). 

This definitely catches Holmes’s attention, and leads him to realize that the assistant is involved and using the cellar in some way. 

The Strange Hours and Activities

For four hours a day, Mr. Wilson sits in an empty office and copies the encyclopedia.  Then one day he gets a note saying that the league is dissolved.  When he investigates he realizes that the tenant was a solicitor and he only rented the room on a temporary basis and there is no way to contact him. 

Holmes realizes that they need Mr. Wilson out of the shop for four hours a day.  They must be doing something there, probably in the cellar.  From this Holmes deduces that whatever the assistant has planned, it is happening soon.

The Assistant’s Pants

It’s time for a visit!  Holmes and Watson go to check out the shop.  Holmes asks the assistant for directions, and then declares that he is the “fourth smartest man in London” and he has “known something of him before” p. 13).  Holmes notices that the assistant has dirty knees, and concludes that he has been kneeling somewhere.  He beats the pavement and notices it is hollow.  Holmes tells Watson he knows something is up.

“A considerable crime is in contemplation. I have every reason to believe that we shall be in time to stop it.  But to-day being Saturday rather complicates matters. I shall want your help to-night.” (p. 14)

Holmes recognizes the assistant and realizes the importance of the dirt on his knees.  He checks for tunnels and finds one.  He knows they are tunneling into the building next door.  Since it is a bank they are after and it’s Saturday, they must act at once.

Holmes arranges for the police (and Watson) to help, and they catch the bank robbers in the act.  

Holmes uses deduction and footwork to solve the mystery of the Red-Headed League.  At first, it does not seem as though a crime has been committed at all.  The League is so ridiculous, though, that Holmes is suspicious.  He knows that there is something there if he digs a little deeper.

Holmes has his own methods, which often involve thinking.  First, he smokes a pipe for fifty minutes, and then he goes to a German music concert to “introspect” (p. 12).  Then he goes to the client’s shop and meets the mysterious clerk, whom he recognizes as “the fourth smartest man in London” and the third most daring (p. 13).

Holmes observes the knees of his trousers, and deduces that he has been kneeling.  He tests the theory by tapping the sidewalk with his stick.  He concludes that the suspect has been digging.  He realizes that since there is a bank next door to the shop, the bank is about to be robbed.

Holmes arranges for the police to lie in wait for the suspects to break through, and he also waits (along with Watson), to apprehend the criminals.

“There are three men waiting for him at the door,” said Holmes.
“Oh, indeed! You seem to have done the thing very completely. I must compliment you.” (p. 17)

In the end, the case is rather simple.  The criminal James Clay simply concoted a scheme to get the client out of the shop so he could break into the bank.

There are many clues. Holmes determines the identity of Wilson's helper through his description:

“Small, stout-built, very quick in his ways, no hair on his face, though he's not short of thirty. Has a white splash of acid upon his forehead.”

and confirms it through his ears:

" Holmes sat up in his chair in considerable excitement. “I thought as much,” said he. “Have you ever observed that his ears are pierced for ear-rings?”

When he visits the shop, he calls Watson's attention to “The knees of his trousers" which are dirty.

The bank being directly nearby is another, and the time the master criminal gets Wilson away from work is a fifth. (There are others.)

The answer to your question can be found in detail in the last few pages of the story. It is characteristic of many of Sir Arthur Conan Doyle's stories about Sherlock Holmes that the master detective explains his solutions to the mysteries at the end, when he and his friend Dr. Watson are back in Baker Street. The following excerpt from "The Red-Headed League" is illustrative of the conclusions of many of the Sherlock Holmes tales.

"You see, Watson," he exclaimed in the early hours of the morning as we sat over a glass of whisky and soda in Baker Street, "it was perfectly obvious from the first that the only possible object of this rather fantastic business of the advertisement of the League, and the copying of the 'Encyclopaedia,' must be to get this not over-bright pawnbroker out of the way for a number of hours every day."

Holmes goes on to explain how he guessed what the motive was. He thought about the assistant's practice of going down to the cellar on the pretext of developing photographs and guessed that he must be digging a tunnel. When he stopped by the pawnshop on the pretext of asking for directions, he noted that John Clay's trousers were "worn, wrinkled, and stained." This proved to Holmes that his guess was correct.

When he examined the neighborhood he noted that there was a branch of the City and Suburban Bank close by. He made inquiries and learned that there was a big shipment of French gold coins being temporarily stored in the basement of the bank.

"Well, when they closed their League offices that was a sign that they cared no longer about Mr. Jabez Wilson's presence--in other words, that they had completed their tunnel."

Holmes knew the burglars would have to act quickly because the bullion might be moved elsewhere. He expected them to make their move on that Saturday night because that would give them all day Sunday to make their getaway. (It would have been too bad for Jabez Wilson if he had happened to drop by his shop unexpectedly. Clay would have had no hesitation about murdering him.)

At the conclusion of Holmes' monologue, Watson exclaims:

"You reasoned it out beautifully. . . . It is so long a chain, and yet every link rings true."

You can refer to the last few pages of the story to read a detailed explanation of the solution to the mystery in Holmes' own words in just four paragraphs. Doyle's creation of Sherlock Holmes was inspired, as he readily acknowledged, by some of the tales of "ratiocination" of the great Edgar Allan Poe. It can be observed that Poe has his hero explain his investigations, observations, and deductions at the end of such stories as "The Murders in the Rue Morgue," "The Purloined Letter," and "The Gold Bug." As previously mentioned, Doyle followed this narrative practice in many of his Sherlock Holmes stories, including "The Red-Headed League."

"The Red-Headed League" is a fairly typical Sherlock Holmes tale in which the narrator, Dr. Watson, sees everything the great detective sees but does not understand his friend's deductions, conclusions, or many of his actions until the end, at which point Holmes explains everything in detail. This explanation occurs in the last few pages of the story, beginning with these words:

"You see, Watson," he explained in the early hours of the morning as we sat over a glass of whisky and soda in Baker Street, "it was perfectly obvious from the first that the only possible object of this rather fantastic business of the advertisement of the League, and the copying of the 'Encyclopaedia,' must be to get this not over-bright pawnbroker out of the way for a number of hours every day."

The reader has been mystified along with Watson up to this point and is happy to learn about the detective's thought processes after having participated in imagination in the initial interview with Wilson, the resulting investigation, and the dramatic arrest of the master criminal.

Holmes explains that the assistant's fondness for photography made him suspect that he was up to something in the cellar. He made inquiries about the assistant and found that he was dealing with "one of the coolest and most daring criminals in London." The only thing he could imagine John Clay doing in the cellar was digging a tunnel. When he and Watson walked around the corner and saw a branch of the City and Suburban Bank, Holmes was sure that Clay was digging a tunnel from the pawnbroker's shop into the basement of that bank.

One of the most striking clues was the condition of "Spaulding's" trousers. Holmes called at the shop pretending to want directions to the Strand and observed that the assistant's trousers were "worn, wrinkled, and stained."

Watson asks:

"And how could you tell that they would make their attempt to-night?"

Holmes replies:

"Well, when they closed their League offices that was a sign that they cared no longer about Mr. Jabez Wilson's presence--in other words, that they had completed their tunnel. But it was essential that they should use it soon, as it might be discovered, or the bullion might be removed."

The burglars chose Saturday night because they knew Wilson would be gone all day Sunday, and they would need a lot of time to move the gold through the tunnel into the pawnbroker's shop, then get a horse and wagon and load all those boxes as discreetly as possible and make their getaway. No doubt they would do the loading in broad daylight rather than trying to be secretive. After all, "Spaulding" was an employee of the pawnbroker and would be known to the neighbors and beat policeman.

When the Holmes and Watson are waiting with three other men in the dark vault for Clay and his partner to arrive, the bank director Mr. Merryweather explains that there are 30,000 napoleons in the crates.

"The crate upon which I sit contains 2,000 napoleons packed between layers of lead foil."

This means that there would be fifteen very heavy crates of gold to be dragged through the tunnel by the two men crawling on their hands and knees, with one of them pushing and the other pulling. The distance from the bank to the pawn shop was considerable. So Holmes was right in calculating that the burglars would almost certainly have to make their move on that Saturday night.

Ah, a fun question.
First, there is the fact that the league itself disappeared. This tips Jabez Wilson off that something is wrong, and leads to the second clue: the existence of the league itself, which is a bizarre thing.

Now, once Holmes is on the case, quite a few clues are spied that Holmes relates to his overall knowledge of London (criminal) society and human nature in general.

Wilson's assistant is too smart for his position; why does he keep it?
His assistant pointed out the position to him; why?
The strict rules keeping Wilson in place during his work are a clue; someone wants him anchored in place.
When Holmes hears what Vincent Spaulding looks like, he recognizes him. This is a major clue, since he recognizes him as John Clay, a criminal.
Spaulding's knees are dirty, indicating he'd been digging.
Then there's the location of Wilson's shop, which is right against a bank.
Holmes thumps his stick against the stones and hears a hollow underneath.

Together, he deduces what they are up to.

Greg

First of all, the whole idea of the Red-Headed League was obviously to keep the pawnbroker out of the office for a number of hours each day.  He could not leave the little office they placed him, or he would forfeit the job.  Also, he was doing this menial task of copying the Encyclopedia Britannica, something that has no financial worth. They gave him an incentive with four pounds a day for doing basically nothing. 

The assistant to the pawnbroker told him that he could take care of the business and would work for half-wages.  That gave Sherlock the idea that he had a strong motive for getting his boss out of the pawn shop for a few hours each day.

The assistant was fond of photography, and he would vanish into the cellar to develop his photographs. Sherlock figured he was doing something down in the cellar that took many hours a week. Sherlock could think of nothing he would do in the cellar for that period of time except dig a tunnel to another building.

He looked into who this assistant was and discovered that he was one of the most daring criminals in London.

Sherlock beat upon the ground with his stick to see which way the tunnel was headed.

He then knocked on the door and the assistant answered.  The knees of the assistant's trousers were worn, dirty, and wrinkled.  That was the precise condition he was looking for if the man were digging a tunnel.

Now he had to figure out which building they were tunneling to --so he walked around the neighborhood and found  the City and Suburban Bank on the property next to the pawnbroker's.

Finally he had to figure out when they were going to do the job.  When they closed the office of the Red-Headed League so quickly, it signaled to Holmes that the tunnel was finished. They had to use it soon or the tunnel might be found and the bullion moved.  So, he figured it would be Saturday night.  That way they would have two days to make their escape.

Arthur Conan Doyle freely acknowledged that he had gotten his inspiration for the Sherlock Holmes mysteries from Edgar Allan Poe's tales of ratiocination about C. Auguste Dupin, particularly from "The Purloined Letter" and "The Murders in the Rue Morgue." Poe uses a narrator who describes what he observes the detective doing but without understanding why. Then the explanation of the protagonist's reasoning process is given by Dupin himself at the  end of the tale. The narrator participates in this unraveling by asking questions that have perplexed the reader.

In "The Red-Headed League," after John Clay has been arrested and carried off to prison, Doyle has Watson quote Holmes as he explains his whole line of reasoning. Adhering to the plot formula invented by Poe, Doyle has the two friends alone in a comfortable room with all the time in the world for conversation. By putting the deductions at the end of the story, the author saves the most interesting part for last, because the reader wants to know how on earth the brilliant detective was able to comprehend what is still a dark mystery to both the reader and the narrator.

"You see, Watson," he explained in the early hours of the morning as we sat over a glass of whisky and soda in Baker Street, "it was perfectly obvious from the first that the only possible object of this rather fantastic business of the advertisement of the League, and the copying of the 'Encyclopeaedia,' must be to get this not over-bright pawnbroker out of the way for a number of hours every day."

Watson continues to ask pertinent questions, such as, "But how could you guess what his [John Clay's] motive was?" Most importantly, Watson  asks: "And how could you tell that they would make their attempt to-night?"

The answer to the posted question, In "The Red-Headed League," how did Holmes guess when the crime would beattempted?,is contained in a single paragraph near the end of the story.

"Well, when they closed their League offices that was a sign that they cared no longer about Mr. Jabez Wilson's presence--in other words, that they had completed their tunnel. But it was essential that they should use it soon, as it might be discovered, or the bullion might be removed. Saturday would suit them better than any other day, as it would give them two days for their escape. For all these reasons I expected them to come to-night."

It would have been a big job for the two burglars to move all those heavy boxes of gold through the tunnel to the pawnshop, then to load all the boxes onto a wagon and make their getaway. Most likely, they would have waited until early Sunday morning before loading the wagon. Since John Clay was an employee of Jabez Wilson, he could have given some plausible explanation to a policeman if one had happened to come along while they were in the process of loading the loot.

The only thing that the ingenious John Clay did not foresee was that Jabez Wilson would experience such a strong reaction to losing his lucrative position at the Red-Headed League office that he would go to Sherlock Holmes for advice. Holmes and Dr. Watson often get drawn into sinister and complex cases which start off with someone calling on them about an apparently trivial problem. Holmes had to be characterized as a man whose hyperactive intellect made him easily bored and who would therefore be glad to take on practically any mystery regardless of its apparent triviality or the possibility of collecting an appropriate fee.

The very oddness of the league is a clue in itself. Who would pay people to do this sort of tedious copying--and who would limit their request to those with red hair?

After that, Mr. Wilson's helper's habit of disappearing into the cellar, and the fact that he brought the league up, signal more generally that something is up. Combine that with the fact that Wilson had not heard of the league before, and you've got some oddity. The biggest clue, though, is that Wilson cannot leave during the period: they want him anchored in place.

Later, there are others--the shop's location, the assistant's knees--but early on, those are the main clues.

Your question is fairly vague—various clues appear to Holmes throughout this story. I will try and guide you in terms of identifying them.

Most Sherlock Holmes stories follow a fairly rigid format. As part of this format, Holmes will meet with a client in his rooms at Baker Street and will immediately be able to deduce various things about the client and the case, based upon what he observes. So, in this story, when Jabez Wilson appears in the sitting room, Holmes is able to deduce several things about him:

1. He has done manual labor at some time.
2. He takes snuff.
3. He is a Freemason .
4. He has, at some time, been in China.
5. He has been writing a lot recently.

Watson and the client are suitably impressed, but Holmes points out the clues that led him to these deductions:

1. His right hand is larger than his left, indicating it has been used more to develop the muscles.
2. He can smell the snuff.
3. The client is wearing a pin with the symbol of the Freemasons on it.
4. A tattoo of a fish above the man's right wrist was evidently done in China.
5. The man's right cuff is shiny, and his left sleeve has a worn patch from being rested on a desk.

Later in the story, when Holmes has cracked the case of the Red-Headed League, he also explains to Watson some "clues" which led him towards his conclusions. He is less methodical here in explaining his deductive reasoning, but he explains that a clue to Clay's motives was found in the fact that there were no women in the house—Clay therefore wanted something that was outside of the house. The cellar was "the end of this tangled clue." Holmes recognized that there must be a tunnel in the cellar going to another building. Meanwhile, the assistant's knees were "worn, wrinkled and stained" which suggested that he had been "burrowing" for many hours. The presence of the bank next to "our friend's premises" provided the final answer: what were the men tunneling for?