Discuss the different stages of the madness of Hamlet during the play.

Your question assumes that Hamlet actually is mad, and I don't think that's the case at all. The whole thing is just a gigantic ruse to unsettle Claudius—to make him uncomfortable on the throne that he stole from Hamlet's father. Hamlet's mad act—for that's what it is—becomes more intense as the play progresses, but that's mainly because Hamlet uses it as a cover for his own chronic indecision. The longer he delays settling scores with Claudius, the more "mad" he has to appear.
At first, Hamlet's feigned insanity seems quite innocuous: amusing even. When he plays games with Polonius, claiming to see the shapes of animals in the clouds, it's all pretty harmless. The same, however, can't be said of his unhinged outburst against Ophelia later on—"Get thee to a nunnery!"—which destroys the poor young lady's already fragile nervous system, and ostensibly leads to her eventual suicide.
The irony here is that Hamlet's mad act doesn't have the desired effect on his intended target (i.e., Claudius). The wily old king is pretty sure that Hamlet isn't really mad at all but wants to make sure all the same. The real victims are those such as Ophelia and Gertrude, who both end up on the receiving end of vicious tongue-lashings from Hamlet that express not so much madness as hatred and contempt—but which are nonetheless all part of Hamlet's devious ploy to appear insane.