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So.  How crazy was Hamlet exactly?  Given the fact that this is one of the more discussed topics encountered when analyzing Hamlet, I struggle to find a way to present my ideas in a way that is not simply a re-hash of a more intelligent person's ideas.  But here goes anyway.

The short version of all of this is that I think Hamlet is crazy, or at least he is by the end of the play (in the beginning I genuinely believe that he was simply faking it).  My thinking on this is primarily centered on Hamlet's treatment of Ophelia, especially when considered in connection to their previous relationship.  Hamlet and Ophelia had a 'thing' going on, to the extent which Polonius felt it was his fatherly duty to remind Ophelia to not stay with Hamlet in private overlong, it didn't look right.  And yet, if Hamlet was truly sane, his treatment of Ophelia makes no sense at all.  He states:

"Get thee to a nunnery: why wouldst thou be a
breeder of sinners? I am myself indifferent honest;
but yet I could accuse me of such things that it
were better my mother had not borne me: I am very
proud, revengeful, ambitious, with more offences at
my beck than I have thoughts to put them in"

He lashes out against Ophelia, the woman he once loved, perhaps too desperately, and says that she should not only remove herself from his presence, but get to a nunnery.  She ruins not just him, but all men by her presence, such that the world would be a better place.  This is unnecessarily damaging to his relationship with Ophelia, even if one assumes that he is only doing so to deceive Claudius and Polonius.  What's more, even if he did so with the intention of his actions going back to the king, it would have been extremely counterproductive.  Polonius already believes Hamlet to be mad as a result of his previous actions towards Ophelia (Hamlet barging into her room half-clothed and writing love-letters) and that this madness is the problem.  By being hostile to Ophelia, Polonius would be disinclined to believe that Hamlet is truly crazy.  Plus, he's pretty negative about himself, and that doesn't usually bode well.

Of course, his interactions with Ophelia are only the tip of the iceberg, though my favorite tip, as it happens.  One could also remark on the fact that Hamlet assumes Claudius is guilty when he runs out of the play about Gonzago Hamlet has put on.  That doesn't necessarily prove anything (even though it is true), for all Hamlet knew, Claudius could have simply had a bad case of indigestion and had to leave to, ah, relieve, himself.

He stabs a person unknown to him, just because they were hiding behind a curtain in his mother's room, though it did turn out to be Polonius, not the Claudius he was expecting.

I want to believe that Hamlet is sane throughout the novel, he speaks so well and wittily, but it isn't to be.  By the end, he's bat-crap crazy.