In Act I, Horatio trembles and looks pale (after seeing the ghost). While these are signs of fear, they can also be signs of disease.
Horatio describes the way that Rome was disturbed when Julius was about to die, and mentions it was sick.
When Claudius discusses the state, he refers to it as being "disjoint and out of frame," as if the nation were a sick body.
Finally, when Hamlet is (essentially) encouraged to cheer up, he refers to the world as being "rank and gross in nature," as if the entire world was sick now that his father had been killed.