The standard grammatical explanation of this is that it’s a variant of constructions like:
What the reason is is that she’d just returned from Guatemala.
These are quite standardly grammatical, analogous to e.g.
What I know is that capuchins are a kind of monkey.
This construction acts in some respects as a fixed idiom, with slightly different connotations from plain old “The reason is that she’d just returned…”, and as such, it’s started to evolve independently. In particular, it’s developed the variant which omits the what, which occurs frequently enough that descriptive linguists happily accept it as grammatical, though slightly nonstandard.
Those different connotations are subtle; the following is my subjective impression, but if someone can find a proper corpus-based analysis of them, that would be better.
The form “The problem is that I don’t know why he’s angry.” can be the first mention of the fact that there’s a problem; it puts focus on this assertion. Contrastingly, “[What] the problem is, is that I don’t know why he’s angry.” is typically used when the listener/reader is already aware that there’s a problem; it emphasises the delineation of precisely what the problem is, possibly in contrast to other things it could be:
The problem isn’t that they’re stupid. What the problem is, is that they’re overspecialised.
(Mark Liberman also discusses the “The X is, is” construction on Language Log, and partially disagrees with this standard analysis, linking to an alternative proposed explanation. I’ve not read the linked paper, I’m afraid, and the standard analysis makes sense to me, so I’m leaving it at this for now.)
In any normal context, a fatal injury is one which either has already led to death, or appears certain to do so.
There will be rare circumstances where that appearance of impending death turns out to be mistaken, but in retrospect this would constitute a misuse of the word fatal.
The normal term for injuries which may well result in death is life-threatening injuries.
With due regard to @Hackworth, when and if it becomes apparent that a person who suffered a life-threatening injury will in fact survive, that could then be called a near-fatal injury. In loose speech people also use this expression for an injury which is not in fact life-threatening, but which could easily have been so had it been slightly different (for example, a knife-wound very close to the carotid artery).
Best Answer
Your use of the word premise is entirely correct:
As you can see from the examples, the use of "premise" is not limited to books, stories, movies, etc., however in filmmaking
In my opinion, the filmmaking definition of premise boils down to the same thing
EDITED TO ADD:
Your sentence is grammatically sound. I have absolutely no doubts about that, and I reaffirm my answer.
I haven't watched TWD, so I honestly don't know if the premise that "people die and turn into zombies" is, in fact, the premise of TWD, as the ensuing debate in the comments alleges it isn't.