"I will kill you" is a straightforward statement: the speaker will take some action to end the life of the person he's talking to. This may be meant as a literal threat, as an exaggerated way to express anger, or as a joke.
"I will kill you dead" does not make sense. Presumably if you kill someone, you will cause them to be dead, so as best the word "dead" is an unnecessary addition. "Dead" is an adjective, like, "Marley was dead", or "I found a dead body in the field". Here it appears to be being used as an adverb.
It is very rare to use "dead" as an adverb, but it is done in a few stock phrases to distinguish a fatal event from a non-fatal event. Like another poster mentioned the phrase "he was shot dead". We also sometimes say, "he fell dead", meaning he fell to the ground and he was dead, as opposed to simply unconscious or just tripped. I can't think of any other such phrases, but there may be a few. You cannot use "dead" as an adverb in general. You can't say, "He caught the flu dead" or "He was in an accident dead".
I can only think of one time I've ever heard someone say "kill dead" like that. That was in a comedy, where a private is afraid that his sergeant is going to be very angry at him for some mistake he made, and he keeps whining, "he's gonna kill me, he's gonna kill me dead". It was clearly intended to be a humorous way to add emphasis. Not only is he going to kill me, but he's going to kill me DEAD.
I'd take it that way here. The speaker is just trying to emphasize the word "kill".