In "Hamlet", describe the madness that Hamlet displays.
After his father's ghost hints at murder, he displays a variety of loopy behaviors. He starts by deriding Polonius, calling him a variety of insults: "fish-monger", "wrinkled...eyes purging thinck amber...having a plentifiul lack of wit" (II.ii.173-202). He then compares himself to a crab, and says he walks into his grave. Later, he harasses Ophelia, denies he ever loved her, and forbids her to ever have children. He acts overly-excited at the play. He then harasses his mother, accusing her of incest, telling her to open her eyes and be wary. He stabs Polonius and then makes jest of it when they ask where his body is, saying Polonius is at supper where "a certain convocation of politic worms are e'en at him" (IV.iii.18-22).
All of this is bizarre behavior, but was it madness? He did all of these things in front of others, or for a certain effect; he was obsessed with finding the solution to his father's murder, and madness is a convenient ruse for gathering clues and not arousing suspicion as you do so. It was also a nice way to express some of his anger without being held accountable for his harsh actions and words. Unfortunately, the plan goes awry, as his actions incite a chain-reaction that leads to the death of all who he cared for, including himself at the end.