Hamlet is a normal teenager much like the rest only with much bigger problems on his plate. Not only is he the former king's son, but his own mother decided to marry the same guy who killed his dad. Aside from that being an incredibly awkward dinner each night, that would take a serious mental toll on any adolescent. Emotions ranging from sadness to anger and to confusion, Hamlet has a lot going on in his head. It would be very understandable to see Hamlet lose his head and lash out on everyone and everything around him, yet, he manages to keep calm throughout the story and bottle his rage so he can formulate a plot to cleverly avenge his predecessor's death. Hamlet's ability to control his madness and lock it behind the curtains of his poker face is a major reason why he is not the madman most believe him to be.
Some may argue that Hamlet is indeed mad for the fact that he acts so strange to the other characters. For example, when Hamlet speaks to Lord Polonius about his daughter in Act II, Scene II. Hamlet acts so odd that is make Polonius say, "...he is far gone, far gone..." While most would argue that the text here is evidence of Hamlet's madness, Hamlet is merely using the mask of madness as a ploy. When around Horatio, or the actor later on in the play, he is completely normal and charismatic. He wants to make sure all of his enemies' guards are down for when he chooses to initiate his diabolical plan. Another argument could be made about Hamlet's reaction to when he kills Polonius. Rather than being shocked at the sight of a different man bleeding to his death, Hamlet's reaction is completely indifferent. Hamlet is more angry about Polonius' intrusion than frightened by how he just killed a man who he did not intend to. While one would say Hamlet is crazy for committing an act of such lunacy, I argue back that he is simply reacting to his environment and bodily emotions. When Hamlet finally talks to his mother about everything he has been bottling up inside him, he explodes in fury. Adrenalin took over as he spewed out vulgar words of hate and disgust giving his mother the thought that he was going to kill her. Hamlet had been so quiet and calm for so long that when he finally got to vent his emotions, he couldn't hold back, much like many other teenagers will tend to do when holding a certain emotion back for too long. So when Lord Polonius show up out of nowhere, Hamlet doesn't take it as much of a shock to see him fall to the ground after stabbing him to death. His eruption of anger, frustration, and adrenalin numbed his ability to empathize at all with the "wretched, rash, intruding fool."
Hamlet is a man with many problems and issues. His psychology however, is still sound. Although he may be a killer and a devious schemer, his "crafty madness" is what makes him such an intelligent character. He is mad, but he is not a madman.