A lot of this is fluff, but I like the overall point that Hamlet isn't all that mad. He's made a choice to behave that way.

"Crazy"? No. Shakespeare understood human psychology way too well to reduce him to madness (or even melancholy).

Try aligning him with more modern characteristions of mental health and you'll start to understand him better.

Hamlet is 30. Young, perhaps, but not that young. Assuming he is a real person living more or less in the same milieu as Shakespeare himself, it would be hard to believe that he doesn't have a pretty good, first-hand sense of what death was all about. Death was not so mysterious in the Early Modern, after all. His acting out as insane is pretty explicitly to suss out what the ghost says. And it works pretty clearly. In case after case, the observers of the "madness" respond in ways that express their own vision of things. Polonius is focused on his daughter, so it's love-madness. Gertrude is, it seems, still struggling with the family tragedy, so it's daddy's death and mommy's o'er hasty marriage (if I'm right about that, then it follows that she is not in on the murder, by the way). R&G are ambitious courtiers, so it's ambition. In that chain of responses, I'd include Claudius, whose suspicions reflect his culpability.

The rashness of killing Polonius seems to me to reflect a thematic concern of the play, which Hamlet states in almost a thesis-like kind of way when he tells Gertrude that it she "assumes" a virtue, and then performs that virtue consistently, the next thing you know she will be in fact, not simply in performance, as virtuous as she has behaved. So Hamlet "assumes" a madness, performs that madness consistently, and the next thing you know he behaves "rashly," madly. The issue is controverted in the text itself, to be sure. So "'Tis not this inky cloak," which he "assumes" in response to daddy's death, that can "denote me truly," says Hamlet—the performance is not the reality, he asserts, and so contradicts what he says to Gertrude later. I've always thought it kind of odd, though, that if the inky cloak does not truly denote Hamlet, then it's superfluous for him to put it on in the first place. I'll grant that maybe in this case the reality precedes the performance, but if that's the case it becomes, to me anyway, almost impossible to distinguish between the reality and the performance.

Obviously Claudius is the test case. He "assumes" the role of a good governor. As that first speech of his says, he's in control of domestic policy and of foreign policy, and as the talk among the watch in 1.1 indicates, of military policy as well. It's only Hamlet's inky cloak that stands in the way of the performance being the reality.

I think the difference is that Hamlet is still alive to ask for forgiveness, which is a crucial detail when considering sin in a Christian context. Suicide is such a taboo in that culture—more abhorrent even than murder—because the sinner cannot seek absolution. As a side note about Hamlet's speech to Laertes... Despite the fact that he declares his madness, I take the speech as the firmest evidence that he is not mad. Laertes is lying too, so the exchange is more compelling for the audience if it's a lie on both sides.