How do the images of rot, disease, and decay that occur throughout Hamlet serve as extended metaphors for Claudius's evilness?

Perhaps it helps to think of rot, disease and decay as recurring motifs in Hamlet that inform one of the major themes of the play--the poison of humankind's baser nature seeking to contaminate or even destroy his better nature. When Marcellus says early in the play, "Something is rotten in the state of Denmark" (I:4:90), he is saying much more than he knows.

Claudius, of course, is a prime example of one who is infected with this disease and decay that has, in fact, infected most of the Danish court. Consider, for instance, the likes of Gertrude, Polonius and even Hamlet himself.  Nonetheless, it is Claudius' heinous crime driven by his apparent lust for Gertrude and his insatiable hunger for power that epitomizes the rotten and diseased character of human nature. Claudius likely typifies what is worst in humankind, and he recognizes that in a moment of illumination.  He laments, "My offense is rank.  It smells to heaven" (III:3:36).

So, while these motifs do indeed recur throughout the play (consider, e.g., I:4:36-38; III:1:45-47; III:4:42-44; III:4:147-49), they find their most sobering expression in the character of Claudius, his heinous crime and the fruit of that crime, "My crown, my own ambition, and my queen" (III:3:55).