"There is something rotten with the state of Demark." How does the rest of the play disprove or prove this statement?
"Something is rotten in the state of Denmark" is the exact quote, and what Marcellus is talking about when he says this line in Act 1, Scene 4, is the fact that he's just seen Old Hamlet's ghost wandering around, despite the fact he's been dead for several months.
The reason for this is because Claudius' has murdered the king, and usurped the throne, and the "something rotten" at the centre of Denmark is the current king, Claudius. I remember reading a critic (though I can't for the life of me remember the name) arguing that Claudius was like a maggot in the apple of Denmark, and gradually, gradually (from Rosencrantz and Guildenstern, to Polonius, to Gertrude, to Ophelia, to Laertes) his way of operating - surveillance, sneakiness, underhandednes - spread through the apple. Until, at the end, everyone is dead.
Denmark is usually thought of as a court full of spies, full of people hiding, full of people frightened to speak out. What's rotten then in the state of Denmark? The king, who sits right at the centre of things, and who set off the realpolitik which abounds in the court.