What is the "how" doing in one sense of the sentence "How would you like to die?"
Solution 1:
This is a peculiarity of the verb to like: the adverb how can be used to indicate the degree to which one would like something, which is not the way how is normally used with other verbs. You could say it is short for how much would you like to die?.
This "degree" is mostly merely rhetorical or polite: hypothetical how (much) would you like to be a member of our club? usually means the same as simply would you like to be a member of our club?. It is mainly used with proposals.
Your example how would you like to die? can mean two things, analogous to how would you like to have your breakfast this morning (manner)? and how would you like to come with me (rhetorical degree)?
How much would you like to die, i.e. would you like to die? [usually a rhetorical question, which is really an announcement rather than a question]
In what manner would you like to die?
Solution 2:
I have not seen the episode in question, so I may be well off the mark, but I do not see how "how would you like to die" could be interpreted as simply "would you like to die", at least if the character is in fear for his life. They are two different questions; the first assumes imminent death and affords the option of method, while the second is a simple challenge. It's (again, to me) the difference between "how shall I kill you" and "shall I kill you".
Solution 3:
I agree with PSU: the question How would you like to die? is in no case synonymous with Would you like to die? The humor does not arise because the answerer misunderstands the question, but because he deliberately ignores the context in which the question is asked.
That is, when A asks B How would you like to die?, he presumes that B is going to die very soon, probably at A's hand. However, B's answer subverts that assumption, as B states that he'd really prefer to die a long time from now, in pleasant circumstances that have nothing to do with A.