Claudius tries to find out all sorts of information about Hamlet by sending spies--it seems to be his chief means of information.

But I just noticed last week that in the two "big" eavesdropping scenes (the ones where Claudius and Polonius listen to Ophelia and Hamlet's "nunnery" scene, and the one in Gertrude's closet when Polonius is killed), it is Polonius, and not Claudius, who makes the suggestion that they participate in trying to overhear the truth.