I think there are 3 parts to your question.
I'll be dead before", implies you will never do something or something will never happen.
"Answer to", means you will take orders or explain your actions to someone.
"To the likes of you", implies somebody you do not respect for whatever reason in that context.
So in total, "I will never explain my actions or take orders from you because I don't respect you"
I'm sure there will be others who will be able to give a better technical definition.