The YES, HE WAS MAD side: Hamlet appears to act mad when he hears of his father's murder. At the time he speaks “wild and whirling words” that appear senseless to Horatio and Marcellus [Act I, Scene v, lines 127-134].

Did Hamlet actually go insane?

Hamlet says in asides throughout the play that he is not, in fact insane, but sometimes his performance is so convincing that it is difficult to tell. Indeed, Hamlet himself wonders if he is not mad, most notably when he sees his father's apparition in his mother's bedroom.

Does Hamlet suffer from madness?

His madness and melancholy has driven him to the point wanting to commit suicide. Hamlet's madness likely stems from an actual mental illness, most likely a depressive illness. Hamlet admits to suffering from melancholy. The death of his father only worsened a preexisting condition.