Hamlet himself, in the play by William Shakespeare, seems to be asking himself the same question in his speech about whether it is nobler in the mind 'to suffer the slings and arrows of outrageous fortune' or to get out there, do something about it and exorcize any crazy ill effects of the injustice by wreaking revenge. Even today, many therapists recommend that we do not internalize or 'bottle up' resentful angry feelings which arise from being victims of unfair treatment. However, I think they would recommend Hamlet to 'talk it through' rather than to go around murdering people. He lets it all get to him, partly through the grief of berffeavement and partly to fulfil public expectations of him and what is considered the noble thing to do for a person of his status.
First of all, I do not believe that Hamlet is truly crazy. I think that he has emotional problems and tends to be depressed, but I do not believe that he is crazy.
As for why he might seem crazy, I would say it is a combination of his personality and the situation he finds himself in. Hamlet is more of a thinker than a man of action. Unfortunately, he finds himself in a situation that demands action. He feels that he really ought to take revenge for his father's death, but he is not really naturally inclined to go out and do it. This discrepancy between what he thinks he should do and what he is naturally inclined to do makes him have emotional problems during the play.
In Shakespeare's play Hamlet is pretending to be mad/crazy. He has good reason to feel crazy. His father's ghost is seeking vengeance for his murder. Hamlet's mother married his father's murder. Hamlet may be interested in his mother in an unsonly manner.
Hamlet is developing a plan to obtain revenge for his father's death. By pretending that he is mad, he is making himself look like he is not a threat. He does talk to himself a lot, but that is because he is moody and debating his own actions. His situation is a difficult one to resolve.