By the time the particular murder scene to which you refer occurs in Macbeth by William Shakespeare, Macbeth has already developed a taste for murder. He is a soldier, he is a bloody assassin, and in Act III he hires mercenaries to murder his best friend Banquo. It is the murder of Banquo which makes us wonder about a mystery murderer.
First of all, King Macbeth has to get rid of Banquo, for he is the only person other than Lady Macbeth who knows about the witches' prophecies and could suspect Macbeth of killing Duncan for his own advancement. He calls two murderers to the castle and, after he feeds them full of Banquo-hate, hires them to do the job.
In Act I scene i, Macbeth makes the arrangements with two murderers. The actual incident takes place in scene iii, and suddenly three murderers appear. This mystery murderer is only called "Third Murderer," and he is the one to whom you refer. Though there is no way to be sure who it is, but we do have a few clues and can make an educated guess.
The original murderers are surprised to see him and ask who sent him. He says Macbeth sent him. They are a bit disgruntled and offended, stating that Macbeth had no reason to distrust them since he gave them clear instructions and should have trusted them to carry them out.
The Third Murderer does not speak much, but everything he says demonstrates particular knowledge about Banquo and his habits. He also exhibits a special kind of urgency to make sure that Banquo and his son, Fleance, do not survive.
The Third Murderer is the one who interrupts one of the others to tell them to listen because he hears horses. Next he tells them the next thing Banquo is likely to do:
he does usually—
So all men do—from hence to the palace gate
Make it their walk.
Clearly he knows Banquo's habits as well as those of other riders from the castle.
When Banquo and Fleance enter with a torch, it is Macbeth who identitifes Banquo, saying, "It is he."
The murderers kill Banquo but Fleance gets away. One of the murderers kind of suggests that this is okay because it was Banquo they were hired to kill. The Third Murderer then says:
There's but one down; the son is fled.
He cares. We hear no more from the mystery murderer.
Again, though we cannot know for certain the identity of murderer number three, we can surmise that it was Macbeth himself. It is unlikely that Macbeth would have hired a third assassin; why wouldn't he have hired them all at the same time and then given them all the same directives. He is often played by Macbeth in stage and movie productions, as well.
The mystery man obviously knew more about the habits of the victim and the castle than the other two murderers, he is the one who identified Banquo by sight, and he is the most upset of the three when Fleance gets away (remember Banquo's prophecy and you will know why Fleance is a threat).
Also consider Macbeth's mindset at this time. He has killed Duncan and his guards with his own hands, and he has been experiencing hallucinations and bouts of paranoia since then. He is not a sane man, and it would be in his own best interest to make sure these murders are carried out as he ordered.
If you haven't read the scene iv yet, you may not want to read this. We know that Banquo's head was smashed with twenty sharp blows, probably making him barely recognizable. Yet, when Banquo's ghost shows up at the banquet, Macbeth recognizes him immediately, perhaps because he saw his bloody body lying in a ditch after he helped murder him.