One of the most powerful signs we are given is after Hamlet's interaction with the Ghost. The soldiers that are with him and Horatio are worried that Hamlet is mad as he pursues the ghost into the dark of the night, concerned both for his safety and his sanity. They think him perhaps possessed by some madness and chase after him with the hopes of rescuing him from himself.
One might also take his response to the Queen and Claudius' questions and discussing his incredible grief as a sign of madness. Everyone is rejoicing in the marraige of Claudius and the Queen and Hamlet is still wearing black and moping about the place, seemingly unable to let go.
Many theories have been advanced with regard to Hamlet's "madness". The most convincing view in this respect, however, is that his madness is feigned, because Hamlet acts normally when he chooses to, and in the presence of those with whom it is safe to do so. Besides, we should not forget that, after his talk with the ghost in Act I, he had forewarned horatio of his intention to "put an antic disposition on". On certain occasions he does appear to be almost crazy, as when he murders Polonius or when he leaps into Ophelia's grave. But even such behaviour can be explained as the result of an excess of bitterness and melancholy, and not indicative of madness. No madman would talk in his soliloquies as Hamlet does.