What is Hamlet's fatal flaw?

Traditionally, the hero's fall in tragedy is attributed to what is called a "tragic flaw." This flaw is some fault of character such as inordinate ambition, jealousy, quickness to anger, or excessive pride. For Hamlet, it is mainly his inaction, rather than action or emotion, which contributes to his tragic flaw. Hamlet's tragic flaw is his indecisiveness and irrationality. 
When Hamlet first encounters his father's ghost and is told that he was murdered by his brother, Claudius, Hamlet is enraged. But rather than acting, Hamlet is determined to learn the truth before he acts against Claudius. So, when a troop of actors comes to Elsinore, Hamlet has the actors perform a scene that reenacts the murder of King Hamlet. At this point in the play, Hamlet and his friend Horatio observe Claudius leap up and leave the room. Since Horatio agrees that this proves Claudius's guilt, Hamlet goes to kill Claudius. However, he fails to act when he discovers that his uncle is praying because he does not want to make Claudius a martyr.

Now might I do it pat. Now he is a-praying.
And now I’ll do ’t. And so he goes to heaven.
And so am I revenged.—That would be scanned.
A villain kills my father, and, for that,
I, his sole son, do this same villain send
To heaven.
Oh, this is hire and salary, not revenge. (3.3.74-80)
 

Hamlet's indecisiveness causes him to procrastinate. He engages in much self-debate in his soliloquies and, after upbraiding himself in these soliloquies, Hamlet sometimes acts; however, his actions are often irrational. For instance, when he goes to his mother's chamber, he frightens her with his bizarre behavior, and then he impulsively assumes Claudius is hiding there and stabs Polonius to death behind the Arras without saying anything. 
 
In Act V, Hamlet tells his friend Horatio that he has a "war in my brain." When he is indecisive about dueling with Laertes, Hamlet yet engages in a debate:
 

I shall win at the odds...
We defy augury. There’s a special providence in the fall of a sparrow. If it be now, ’tis not to come. If it be not to come, it will be now. If it be not now, yet it will come—the readiness is all. Since no man of aught he leaves knows, what is ’t to leave betimes? Let be. (5.2.204-210)
 

Hamlet is a complex character who possesses a paralyzing intellect and a destructive irrationality.