Is Hamlet insane?

There are two ways to answer this (I guess that goes without saying): either he is most certainly insane, and acting in insane ways during the play, or not insane, but acting in calculated ways in order to achieve certain goals. 

If you argue that insanity is what guides Hamlet, then we can see that as soon as his father dies, he speaks with "wild and whirling words" [Act I, Scene v, lines 127-134], indicating a sort of madness. His mood shifts easily throughout the play, as do his affections and behavior to Ophelia. He kills Rosencrantz and Guildenstern, although they aren't directly involved in the scheme to avenge his father's death. Hamlet also tells Laertes that he killed Polonius "in a fit of madness" [Act V, Scene ii, lines 236-250], and then won't divulge where he put the body. He talks to a skull, for crying out loud! It's pretty damning evidence for insanity.

If you argue that he was not insane, but merely feigning it, then you can see that, in Act i, Scene v., lines 166-180, Hamlet tells Horatio that he will "feign madness", and to excuse strange behavior from him. Hamlet only shows strange behavior around certain characters, as well, including Polonius, Rosencrantz and Guildenstern, Ophelia, and Claudius and Gertude. He acts rationally around Horatio and the Gravediggers. Most notably, we can contrast Hamlet's behavior to Ophelia's behavior, and we can tell the difference between feigned insanity and real insanity.

I think from one perspective it certainly seems as if Hamlet was progressively going mad throughout the play, from seeing ghosts, to talking to a dead man's skull, to killing his uncle and mother. I personally think he was an incredibly smart man, biding his time (or being indecisive, as others argue) and manipulating others until he could fulfill his promise to avenge his father's death. I don't believe for a second that he was insane.