Hamlet is sane, despite the fact that he appears to be insane on the outside, as evidenced by his capacity to look into his father's murder, learn new knowledge, and plan against other characters in the book.
Hamlet is sane because, throughout the play, he only exhibits irrational behavior in front of a select group of people. Around the rest of the group, he behaves properly, acting like a proper prince who can deal with them without coming off as crazy. Finally, despite everything that has happened in his life, he is still able to exact revenge. Throughout the entire play, Hamlet maintains his sanity and deceives Claudius and his cronies by feigning madness.
Hamlet never exhibits any symptoms of true insanity in the course of the play; instead, he merely creates a fake impression of insanity to further his quest for vengeance. Hamlet alternates between sanity and insanity throughout the play, whether through acting crazy to annoy those he doesn't like or to keep himself out of danger. The reason Hamlet can pull off acting crazy so successfully is because he is one of the play's sharpest characters.
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