There is one definite connection between Death Wish and Hamlet. The tool of vengence is not a perfect tool of justice. 

As pointed out above, the character of Hamlet is destroyed and consumed in his role as the the Figure of Revenge. He is not "good" or innocent. This is true of Bronson as well. However righteous his rage may be, Bronson's character chooses to cross the line and defeat evil with evil. 

We might argue that justice is done in both stories, but this justice is not carried out by a white knight but by a sullied individual who, so sullied, has nothing left to lose morally. 

I believe it is also a conscious reaction to the gratuitous violence of Elizabethan revenge drama, which tended to portray murder graphically on stage. (Rather like Charles Bronson, perhaps.)

I do not think Hamlet gives us the vicarious pleasure of seeing a terrible deed avenged. Claudius indeed dies at Hamlet's hands at the end of the play, but not until the chain of events he sets in motion with the murder of Hamlet's father has destroyed every major character in the play. We also see throughout the play that Hamlet's desire for revenge consumes him--in the end, it kills him both physically, and, we might argue by that point, mentally as well. So I don't think the two compare all that well, as Shakespeare delves far deeper into exploring the ambiguities and costs of revenge than does the Bronson film (as I remember it.)