the end of the penultimate paragraph of the story. The narrator's tension is clear, adding to our own when he says:

Oh God! what could I do? I foamed -- I raved -- I swore! [...] but the noise arose over all and continually increased. It grew louder -- louder -- louder! [....] I felt that I must scream or die! and now -- again! -- hark! louder! louder! louder! louder!

In this moment, just prior to his confession, the story reaches its climax.

This is one of Poe’s most suspenseful short stories.  Like much of this story, the suspense in the climax is psychological.  The story basically centers on the narrator, who is struggling with insanity.  He kills the old man, his roommate, because of his evil eye.

The climax of the story is not when the narrator kills the old man, cuts him up, and buries him under the floorboards.  Instead, the climax is when the narrator is driven mad by the old man’s still-beating heart, which he can hear in his mind.

The climax of the story is suspenseful because the police come and the start looking around, and the narrator makes small talk and tries to be casual and clever.  He even has the policemen sit in the same room where the body is buried!  Alas, he is driven so mad by the imaginary beating heart that he confesses.

The narrator begins to think about killing the old man he lives with. Gradually he becomes obsessed with his desire to kill him although he doesn't really understand why he wants to do so. He keeps spying on the old man while he is asleep. The reader knows that sooner or later the narrator, who is obviously a madman, is going to commit a horrible murder. The suspense keeps building. The reader knows that there is going to be a murder committed but does not know when or how it will occur. Even the old man seems to know his life is in danger. He lies there in fear each night but seems paralyzed and unable to take any action to save himself.

When I had waited a long time, very patiently, without hearing him lie down, I resolved to open a little—a very, very little crevice in the lantern. So I opened it—you cannot imagine how stealthily, stealthily—until, at length a simple dim ray, like the thread of the spider, shot from out the crevice and fell full upon the vulture eye.

The narrator begins to hear what he takes to be the sound of the old man's heart beating furiously because of his intense fear. This heartbeat will continue to be a strong effect throughout the rest of the story and will give the story its title. 

But even yet I refrained and kept still. I scarcely breathed. I held the lantern motionless. I tried how steadily I could maintain the ray upon the eye. Meantime the hellish tattoo of the heart increased. 

And then, finally, the narrator acts.

The old man's hour had come! With a loud yell, I threw open the lantern and leaped into the room. He shrieked once—once only. In an instant I dragged him to the floor, and pulled the heavy bed over him.

For a brief interval the narrator thinks he has committed the perfect crime. He dismembers the body and hides it under the floor. But the old man's single shriek and the murderer's "loud yell" have alarmed the neighbors. The police show up at four o'clock in the morning, just after the narrator has put everything back in order. The narrator behaves with complete composure, and it appears that he is going to get away with the murder. After all, everything looks perfectly normal. But then he begins to hear that beating heart again. He believes it is his victim's heartbeat, even though he has cut off the old man's head as well as his arms and legs. The heartbeat continues to drum louder and louder in his ears. Evidently it is his own heart beating with fear of being exposed as a murderer.

Yet the sound increased—and what could I do? It was a low, dull, quick sound—much such a sound as a watch makes when enveloped in cotton. I gasped for breath—and yet the officers heard it not. I talked more quickly—more vehemently; but the noise steadily increased. 

The sound is so obvious to the narrator that he believes the policemen must also be well aware of it. If so, why do they say nothing about it? Why don't they seem suspicious? The narrator assumes they are playing cat-and-mouse with him. They are only waiting for that tell-tale heart to make him break down and confess. And finally he does so.

“Villains!” I shrieked, “dissemble no more! I admit the deed!—tear up the planks! here, here!—It is the beating of his hideous heart!”

He must be completely insane to believe that the old man's heart could still be beating after he had murdered him and cut his body into pieces. There has been a rising feeling of suspense since the beginning of the story. The tension increased up to the time of the murder. Then it began to increase even further when the police arrived. The narrator was not in any danger before the arrival of the police. The murder only made the tension greater because it brought the forces of the law. The narrator's exposure illustrates the thesis of Poe's story, which can be stated in the words used by Hamlet in Shakespeare's play:

For murder, though it have no tongue, will speak
With most miraculous organ

Macbeth expresses the same idea in another Shakespeare play:

It will have blood: they say blood will have blood.
Stones have been known to move and trees to speak;
Augures and understood relations have
By maggot pies and choughs and rooks brought forth
The secret'st man of blood. 

Simply put, the old man was killed by the narrator. The first line "True!—nervous—very, very nervous I had been and am; but why will you say that I am mad?" alerts the reader that the speaker is clearly unreliable, and we are alerted to the fact that something unpleasant has occurred. Any act that would label someone as "mad" for having committed can't be good news. The narrator goes on to tell us that the murder was carefully planned beforehand, & it seems to have been sparked by one trivial physical trait: the cataract over the old man's eye. For an unknown reason, the old man’s cloudy, pale blue eye has incited madness in the narrator. He believes that only by killing the old man can he get rid of the eye’s overpowering malignant force.

Thus, he plans out his crime. For a whole week, he enters the old man's room at night, but the victim remains sleeping each time. The narrator cannot bring himself to kill the man without seeing that which drives him into madness, so he continues entering the room until the old man wakes. On the eighth night, he wakes up and cries ‘‘Who’s there?’’ The narrator does not respond, but instead waits silently for an hour. Eventually, the narrator shines his lamp on the old man’s eye. Heis overcome with madness at the ‘‘damned spot,’’ but he soon hears the beating of a heart so loud that he fears the neighbors will hear it. With a yell, he leaps into the room and kills the old man.

Near the end of the story, it appears that the narrator is going to get away with his crime of murdering the old man. He is welcoming to the police and never loses his cool as they question him and search for the body. The police don't find the body. The narrator is even bold enough to put chairs for the police on the floor right over the spot where he has buried the old man. For all intents and purposes, the narrator has succeeded in deceiving the authorities.

The surprise ending is that the narrator is the cause of his own downfall. As he sits with the police, he thinks he hears the heart of the murdered man beating louder and louder. He also thinks the police detectives hear the sound too and are playing a joke on him by pretending not to. The sound of the heartbeat and the conviction that the police are toying with him are both so overwhelming to him that he confesses to the crime.

The surprise ending comes when our main character is so plagued by the beating of his own heart, which he perceives to be the murder victim's, that he tears up the floorboards to expose his crime to the law enforcement officers on scene. Throughout the story, our protagonist battles with the beating heart, until at last he can stand no more, and reveals the dismembered body he has concealed beneath the planks of his own floor. As we gain a perspective inside our character's mind, we get to experience a small piece of his insanity. Poe was good at using the first person to engross his readership.

Poe does not reveal much information regarding the character of the old man, who is violently murdered and dismembered by the mentally unstable narrator. The reader learns that the old man likely lives with the narrator, has a seemingly positive relationship with him, and is relatively wealthy, because the narrator states that he has "no desire . . . for his gold." However, Poe does not elaborate on the specifics of their relationship. The reader does not know if the old man is a relative of the narrator or simply his roommate. It is also possible that the narrator is the old man's caretaker or a friend of the old man's family.

The reader also discovers that the old man has a pale blue eye with a film over it, which the mentally deranged narrator refers to as an "Evil Eye." It is the old man's eye that incites the narrator's rage and influences him to commit the murder. The narrator also describes the old man as friendly and respectful, saying,

I loved the old man. He had never wronged me. He had never given me insult.

The old man is also extremely vulnerable and defenseless against his attacker. On the eighth night that the narrator peeks his head into the old man's room, the old man sits up in his bed and senses the narrator's presence. Tragically, the old man cannot defend himself against his attacker and dies a violent death.

We know that the old man had a good relationship with the narrator at one time because as the narrator says,

I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire.

Not much is known about the old man with certainty. Although not the typical interpretation, it's possible that the old man is the narrator's father. It seems as though the narrator and the old man have had a relationship for a while (given the narrator's comments), and they live together (something that might be considered unusual if they weren't family). The narrator also mentions that he had no desire for the old man's gold, which, perhaps, he would have stood to inherit upon the old man's death. Plus, the old man seems old enough to have a grown son.

We do know that the old man likely had cataracts, which caused the "veiled" and pale the appearance of his eye. It is, evidently, the "vulture" look created by the cataracts that so incensed the narrator.

It seems like the old man did not suspect that he was in any kind of danger from the narrator, because he slept soundly for the first seven nights that the narrator intruded into his bedroom. He didn't become alarmed until, on the eighth night, when he heard the click of the lantern and emitted a "low stifled sound that arises from the bottom of the soul when overcharged with awe." It is unclear whether the old man would have suspected the narrator of threatening him on that final night or if he believed the noise was made by something else.

Although the main charactor of the classic Edgar Allan Poe short story, "The Tell-Tale Heart," is a madman and, thus, an unreliable narrator, we are told that he "loved the old man." The old man was apparently wealthy, for the narrator states that he had "no desire... for his gold." The old man must have trusted the narrator, since he allowed him to live in his home. The old man is described as having an evil eye--"the eye of a vulture... all a dull blue, with a hideous veil over it"--and it was his eye that caused the narrator to kill the old man.

The old man always went to bed long before midnight, since it was at that time that the narrator sneaked into his room each night. For a week the narrator looked into the old man's room, but he refused to kill him since the vulture eye was always closed. The old man suspected nothing, since the narrator treated him kindly each day. On the eighth night, the old man heard a noise, and he sat up in bed. When the old man groaned, the narrator knew it was one "of mortal terror." When the murderer finally made his move, the old man "shrieked once--once only." His body was then dismembered and "deposited" beneath the planks of the floor.

"The Tell-Tale Heart" is an interesting story to map because the action continually rises until the very end of the story.  The narrator experiences an increasing sense of "madness" until the end when he cannot take the beating of the heart any longer.  The moment when he decides to tear open the floor is the climax or turning point of the story.  At this time, the narrator is overwhelmed by his guilt from having killed the old man, and he cannot suppress this guilt in the face of the policemen.  So he decides to out himself as a means of dealing with his guilt making this the climax of the story.

I disagree with the above interpretation. Since the climax of a story is typically thought of as the moment of the most tension, I would locate the climax in the second to last and last paragraphs of "The Tell-Tale Heart," when the narrator's growing anxiety, fear, and, perhaps, guilt, seem to be getting the best of him, and he confesses to the crime.

Unlike the moment when he actually kills the old man, a moment not particularly tension-filled, the narrator's certainty that the police officers are aware of his guilt renders the penultimate paragraph particularly tense. He thinks that he hears the beating of the (dead) old man's heart beneath the floorboards, and he is sure that the officers hear it too. In just these paragraphs, the narrator uses twenty-five exclamation points to convey his tension.  However, if the climax -- the narrator's belief that he can hear the dead man's heart and that the police know, and the narrator's eventual confession to the murder -- arrives in the final lines of the story, this means that the story actually has no falling action; it concludes with the climactic confession of the narrator and we are left to imagine what takes place afterward.

In this classic short story by Edgar Allan Poe, the falling action starts with the man's dismembering the old man's body and burying it under the floor boards and continues through the arrival and questioning by the investigators and his imagining being able to hear the still beating heart of the man he killed.  Falling action is typically the event or series of events that immediately follow the climax and lead to the ultimate resolution of the story.  In this case, the resolution is the man spilling out the truth that he is the murderer.  He is driven to this final reveal by his guilt over his what he has done. 

This is one of my favorite stories!

Here are some helpful hints on finding the climax of ANY story. Author's usually write their plots in a typical format. Here it is: 

1. The exposition (the part of the story where you get the background and details about the setting, characters, potential conflicts, etc.).

2. The narrative hook, (where the author grabs your attention).

3. The rising action (this is where the author creates tension and suspense through conflict).

4. The climax (the turning point of the story).

5. The falling action  (where the READER learns about the main character's reaction to the climax.

6. The denouement/wrap-up (the final tying together of the story).

Look at Step 4, the turning point of the story. I think you'll find the the climax is where the main character can no longer tolerate his guilty conscience. That dreadful heart beating in his head, is actually a manifestation of his guilt. He can no longer stand the sound, so he rips open the floor revealing his crime! 

It is important not to confuse the author with the narrator of the story; the narrator is not Poe, and Poe does not commit the dark deeds the narrator describes.  However, if you want to know how Poe, the writer, killed off the old man, the character, in the story, I can answer that. 

The unnamed narrator of the story grows to hate the "vulture eye" of the old man with whom he lives, and so he determines to kill him.  After many nights' watch, the narrator says that he threw open his lantern and leaped into the old man's room, smothering him by pulling the heavy bed on top of him.  Once the old man's heart has stopped beating, the narrator dismembers him in a tub (to catch all the blood), and he buries the pieces of the body beneath the floorboards. 

This question can be answered through a close reading of the text; in it, the narrator describes his struggle as he murders the old man, and what he does with the body afterwards.  I encourage you to take a close look at it and read it for yourself, because it's such a cool story, and the way that Poe writes it is super intense.

To summarize, the narrator opens the door to the room and eventually, shines the lantern onto the old man.  And, it is the old man's creepy eye that falls into the beam of the lantern.  This, along with his perception that he can hear the old man's heartbeat, makes the narrator so incensed that he attacks the old guy.  Poe describes it thus:

"In an instant I dragged him to the floor, and pulled the heavy bed over him."

So, he basically smothers the old man with the mattress.  The narrator mentions how it took quite some time for the old man to be smothered to death, and for his heartbeat to stop beating away.  But, after the deed was done, the narrator takes gruesome means to cover the crime--he chops the old man's body to pieces in the bathtub, and buries the parts under the floorboards of the room he was murdered in.  Pretty horrific, if you ask me, which is probably one of the reasons that for nearly 200 years, this tale has been so popular.  Poe has a talent for writing really interesting narrators who do some pretty fantastical things.   I hope that helped; good luck!

Your original question asked more than one question, so I have had to edit it down to focus on the climax of this terrifying story. Let us remember that the climax of the story comes after the rising action, where the story reaches its high point, or the stage at which the reader is most engaged and excited about what is happening. It comes before the resolution, when the conflict is resolved and we are given our ending.

It is clear then that in this story, the climax actually comes when the policemen come to visit, alerted by a neighbour about the shriek that was heard as the narrator killed the old man. Interestingly, at first, the narrator is completely nonchalant about their arrival:

I smiled - for what had I to fear? I bade the gentlemen welcome. The shriek, I said, was my own in a dream. The old man, I mentioned, was absent in the country. I took my visitors all over the house. I bade them search - search well.

The narrator clearly feels that he has nothing to be worried about, having concealed the body. He talks of the "wild audacity of my perfect triumph" as he revels in the fact that the policemen sit in the same room where he has concealed the corpse. The policemen are obviously satisfied, but stay talking to the narrator, until he begins to hear a sound, which with terror he realises is the sound of the old man's heart, still beating.

It is this point that is the climax, as the sound of the heart becomes ever louder, driving the narrator insane and causing him to confess to the murder that he would have otherwise gotten away with.