Hamlet becomes very revengeful after the appearance of the ghost, what is the tragic self discovery that he experiances along the way?
In Shakespeare's Hamlet, Hamlet's mistake is that he waits too long to take his revenge against Claudius, which leads to the blood bath at the end of the play and the deaths of Polonius and Ophelia during the course of the play.
I'm not exactly sure about your words, tragic self-discovery, though. It's not that the self-discovery is tragic. He delays in taking his revenge and speculates during the play as to why, but any self-discovery he makes doesn't cause the tragedy.
The question is, then, why does he wait, and I'm afraid the answer is ambiguous. Numerous reasons for his waiting have been suggested by critics. Hamlet himself at one point says he's a coward, but then later in the same speech reveals that he's not sure that the Ghost is really the ghost of his father--it could be a demon in disguise sent to cause him to kill an innocent Claudius. Hamlet later says that he thinks too much, and that is a possibility. Others say he was depressed, or mad. Others say he is just incapable of action. A very small minority says he's in love with his mother and can't get over it.
Each reader, then, needs to draw a conclusion based on the evidence in the play. And then, in your case, determine whether or not you think Hamlet realizes his fault.