In Claudius's Denmark, death and decay have become very much the norm. We see them everywhere throughout the play, whether it's in Hamlet's dark humor at the expense of the recently slain Polonius, or the young prince's ruminations on death, such as in the famous “To be, or not to be” speech and his equally famous “Alas, poor Yorick” monologue.
With Claudius on the throne, the very air that Hamlet breathes stinks of decay. To him, it is nothing more than “a foul and pestilent congregation of vapors.” It's highly unlikely that the air was much sweeter when Hamlet's father was on the throne; Hamlet is speaking metaphorically, not literally here.
The point that he's making is that Claudius, the man who murdered his way to the throne by killing old King Hamlet, is polluting Denmark by his very presence. The death of Hamlet's father has led to a general air of moral decay in Denmark which Hamlet, almost uniquely, refuses to breathe in. He may not be perfect, but he stands apart from the moral decay that has been eating away at the kingdom ever since Claudius became king. It is this that makes him, albeit imperfectly, the moral center of the play.