In fact the concern over Hamlet's sanity is foreshadowed in scene iv. When Hamlet tells his friend he will follow the ghost and hear what he has to say, Horatio says, "He waxes desperate with imagination" (I.4.87). However, in Act I scene 5, we see Hamlet's extended conversation with the ghost of his father. In this discussion Hamlet remains emotional but rational. He realizes that what the ghost tells him is probably true. He shows no sign of madness, nor does he after this event. Only Horatio and Marcellus see him after Hamlet's talk with the ghost. He despairs about what he will have to do, but still sounds completely rational: The time is out of joint: O. cursed spite,

That ever I was born to set it right!

If the rest of the play is interpreted believing that Hamlet is feigning madness as it meets his needs, everything he does, including the times when he seems to speak irrationally, make complete sense. The king and queen, Polonius, Rosencranz and Guildenstern all believe him to be mad, and have thought of two reasons he might have become melancholy: the death of his mother followed by her hasty marriage to her husband's brother, viewed as incestuous by both the ghost and Hamlet, and his supposed unrequited love for Ophelia. The stage is set for Hamlet to do what he has to do.

Hamlet wastes no time supporting the idea that he is mad immediately falls into his role as the newly crazed nephew of the king. Later in that scene, Hamlet wanders into Polonius' presence reading...

Polonius, who might otherwise have realized he was quite pointedly being insulted, dismisses Hamlet's words as nonsense from someone who unfortunately has a tenuous grasp on reality.
Finally, Hamlet lays his deception out for Rosencranz and Guildenstern to see, if they are alert enough to see the truth. Hamlet tells them:

I am but mad north-north-west: when the wind is southerly I know a hawk from a handsaw" (2.2.378-379).

He clearly tells them that he is mad when it suits him. In addition, he makes a reference to them: they are from the north. His imagery is that those from the north will be deceived into thinking him mad, while those from the south -- Horatio and Marcellus -- will see that he only feigns madness as a ploy to get to his goal and avenge his father's death.

Hamlet shows himself to be cleverer than those who are working against him. The people who believe him crazy -- King Claudius, his mother Queen Gertrude, Claudius' advisor Polonius, and the king's spies, Rosencranz and Guildenstern, miss the truth not because they are stupid but because Hamlet has fooled them with his hoax. Hamlet is not crazy. He is a man with a plan who has the ability to manipulate others into seeing what they want to see while he prepares to executes his vengeance for his father's murder.