Is it correct to say that the "premise" of the show The walking Dead is that people turn into zombies when they die?

Solution 1:

Your use of the word premise is entirely correct:

an idea or theory on which a statement or action is based:

  • [ + that ] They had started with the premise that all men are created equal.
  • The research project is based on the premise stated earlier.
  • We should start from the premise that circumstances might change.
  • The conclusions you have drawn are based on a false premise.
  • We should work on the premise that this plan will be successful.
  • Could you explain the basic premise of your argument? Your reasoning is based on a misguided premise.

As you can see from the examples, the use of "premise" is not limited to books, stories, movies, etc., however in filmmaking

The premise of a film or screenplay is the initial state of affairs that drives the plot.

Most premises can be expressed very simply, and many films can be identified simply from a short sentence describing the premise.

  • For example: A lonely boy is befriended by an alien;
  • A small town is terrorized by a shark;
  • A small boy sees dead people.

In my opinion, the filmmaking definition of premise boils down to the same thing

An assertion or proposition which forms the basis for a work or theory.

  • ‘the fundamental premise of the report’

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.

Solution 2:

The premise of a film or screenplay is the initial state of affairs that drives the plot.

A "premise", per the definition, is foundational to the plot. There's several problems with proposing that your suggestion is the actual premise to TWD.

It does a poor job of describing -this- story

Your suggestion can be re-written, accurately, as "the premise is that the story is set in this genre". I.e., your suggestion equally describes Zombieland, Shaun of the Dead, 28 days later, TWD, and literally every other story within the larger Zombie-setting. As such, it's not very meaningful to describe the premise in this way.

This story doesn't require your premise to work.

Foundational elements of stories often vary independently of the setting. Romeo and Juliet could be told equally well in Verona, Italy, concerning nobleman and ladies as it can in the Bronx, concerning street gangs as it can with Garden-Gnomes. The premise of Romeo and Juliet persists throughout those versions of the classic tale.

Similarly, TWD's premise has very little to do with "zombies", if anything at all.

Opinion:

It's my opinion that TWD's premise is "Man struggles to adapt to post-apocalyptic environment".