Remember that verbal irony happens when a character says something but means the opposite. Can you find a place where one of the characters says something about another character, but you know that he or she is thinking the complete opposite? Hamlet in particular would be a good character to look at.
Dramatic irony happens when the audience (you, in this case) knows something that a character does not. Given the fact that the king is plotting against Hamlet while Hamlet is plotting to kill the king, there are MANY examples here.
Disease imagery is just that. Find an instance in which a character uses language connected with disease to describe/explain a sitation, an action, or another character. This actually won't be hard, since you just have to skim for diesease words. Once you've found one, you can just work with the text around it.