How has Hamlet lived up to his reputation of being a madman in Shakespeare's Hamlet?

First, it is important to remember that Prince Hamlet has intentionally created his reputation of insanity.  In Act 1 after the ghost of his father places the unsavory obligation on Hamlet to "Revenge his foul and most unnatural murder," Hamlet quickly realizes that he will need to create a diversion to cover up his investigation and his revenge plot against his uncle, King Claudius.

Shakespeare's contemporaries would have understood the gravity of Hamlet's decision.  After all, up through 1601, when Shakespeare wrote Hamlet, and beyond, the mentally ill were still generally put to death via various methods of torture designed to drive the devil out.  At best they might be committed to an insane asylum, such as Bedlam in London.  Yet Hamlet very determinedly spreads his own rumor of madness by at first giving up "all custom of exercise," and failing to attend to  his personal appearance.  He boldly enters Ophelia's chambers in castle Elsinor (something a man just didn't do) and proceeds to put on a show of dishevelment and sorrowful madness.  He knows full well that she will tell the groveling Polonius, who blathers to the king and queen that Hamlet is "mad for love."  Hamlet also makes intentionally snarky comments to others under the guise of insanity, such as in Act 2 when he tells Polonius he wishes the counselor was as honest as a fishmonger.  

Yet Hamlet's feigned insanity quickly escalates beyond mere puns and games in Act 3 when he can barely contain himself during "The Mousetrap," making rude sexual comments to Ophelia and blurting out spoilers about the murder plot.  By the time he kills Polonius in his mother's chambers with little remorse, he is clearly no longer in control of his pretense.  As he tells Gertrude, when one puts on a fake behavior, "That monster, custom, who all sense doth eat, / ...can change the stamp of nature."  Hamlet has put on a frock of madness that he seems unable to throw off.  

After he hides Polonius' body and runs about the castle pretending to play children's games with the guards, there is no longer doubt in Dane's mind that the prince is truly a madman. His own mother is unable to believe his insistence to her that he is sane.  No wonder it is so easy for Claudius to convince his court that he is sending Hamlet to England to "protect" him, considering the consequences for madmen during that time.  Ultimately, Hamlet lives up to his reputation of being a madman by violently committing his revenge on Claudius in front of an uninformed audience. The only one left to vouch for the fallen prince is Horatio, who is not even a Dane.