Is Hamlet insane?
Is Hamlet crazy? That all depends on how the actor plays him.
The interesting problem with madness is that the man who says he's not crazy is questioned because...why would anyone insist they weren't crazy unless they were afraid someone would guess the truth? When Hamlet tells his friends that he will behave oddly and for them to not read into it, he essentially does this very thing. Each time he says that he is only acting crazy, he encounters the problem of a crazy person asserting he isn't crazy. Thus, scholars who insist that he is just playacting because he says he is may be correct, but they may not.
It's possible that he went off the deep end the moment he spoke with his father's Ghost, who told him how he'd been murdered (and the fact that his mother had immediately married the killer--"the funeral baked meats / Did coldly furnish forth the marriage tables"--wouldn't help his mental state). Then Ophelia, urged by her father, rejects his sincere "importunes of love." This is enough to drive anyone to the edge, but Hamlet has sworn to his father's ghost that he will avenge his murder, but he isn't absolutely certain that the ghost tells the truth and isn't some random demon there to drive him to murder.
Another possibility is that Hamlet began by "playing" crazy, but after Ophelia's death, he actually did go off the deep end.