In lines 175-177 of Scene 5, Hamlet tells Horatio and Marcellus that he will pretend to be mad or crazy. This way he can spy on his mother and uncle without them knowing that he is only pretending being crazy.
Hamlet will carry out his revenge using subterfuge. He will gather evidences of his uncle's wrongdoing by bearing witness as a crazy person. If people will think that Hamlet has lost his mind, they will lower their guards and talk freely of their plans discounting Hamlet's presence in their midst.