In Julius Caesar, what is Brutus's hamartia?

Brutus’s tragic flaw is his need to be noble. 

Hamartia is a tragic flaw.  It is the cause of a hero’s downfall.  Tragic flaws can come in many varieties, but Brutus’s tragic flaw is his need for nobility.  It makes him naïve and vulnerable. 

We first see Brutus demonstrating his tragic flaw in the speech he makes before the conspirators gather at his house.  It is an important meeting.  He will meet them all together for the first time, and there they will plot.  Before the meeting, he gives a soliloquy in which he explains that he has nothing personally against Caesar, but Caesar has to die.  For Brutus, everything has to be for the good of Rome.  He is concerned about appearances, and it is this desire to do good and the concern for what others think that will destroy those conspiring against Caesar. 

During the meeting, Brutus is adamant about not killing anyone other than Caesar.  He also speaks beautifully and philosophically about what they are about to do, as if they are not really just planning to kill a man.  Brutus gets so wrapped up in his own earnest visions of how people will perceive them and their motives that he loses track of the bigger picture.  Thus, he spews nonsense like this: 

Let us be sacrificers, but not butchers

… But, alas,

Caesar must bleed for it! And, gentle friends,
Let's kill him boldly, but not wrathfully;
Let's carve him as a dish fit for the gods

Not hew him as a carcass fit for hounds (Act II, Scene i) 

Brutus seems to think there is a noble way to kill Caesar.  He believes the ends justify the means, and that no matter what happens everything will be fine because he is the one in charge.  Brutus honestly thinks the people of Rome will welcome him with open arms as their liberator. 

Practicality has no place in Brutus’s world.  He is blocked by his tragic flaw.  When Cassius tries to talk him out of letting Mark Antony speak at Caesar’s funeral, he refuses, again for noble reasons.

You shall not in your funeral speech blame us,
But speak all good you can devise of Caesar,
And say you do't by our permission;
Else shall you not have any hand at all
About his funeral: and you shall speak
In the same pulpit whereto I am going,
After my speech is ended. (Act III, Scene i)

Of course Mark Antony would agree to that.  Why wouldn’t he?  There is no benefit to Brutus speaking first, and saying that he spoke with Brutus’s permission will just play into Antony’s hands.  He intends to portray Brutus and the others as murderers and oath-breakers.  In fact, he turns Brutus’s tragic flaw against him, referring to him and the others ironically or sarcastically throughout the speech as “honorable men.”

A character’s tragic flaw is usually what dooms him.  Brutus doomed himself and all of the conspirators.  He ended up committing suicide, because it was the only honorable way left to him after he botched everything so badly.  Brutus made every decision based on what he thought was honorable, but in the end his only choice was to die an honorable death by his own hand.