It's valid usage, but it implies you think the show is generic
A premise is what forms the basis of a theory or a plot
The premise is the core idea of a work of fiction - the essence from which everything else follows.
It's a strong word. You can say that the premise of The Walking Dead is that people die and come back as zombies, and it would not be incorrect usage of the word - but it implies criticism if you say the premise is something so thin and generic that it is true of any work of zombie fiction.
"The premise is that people die and come back as zombies" is something a critic might say in a scathing review accusing The Walking Dead of being just another formulaic zombie show.
This is almost certainly why your friend questioned your use of the word. It was probably clear from context that you believe there is more to the show than some people dying then coming back as zombies.
Some real usage examples
Let's compare that to some typical usage of the word in this context:
The premise is that a dangerous criminal has threatened carnage at the Miss United States Pageant
The movie's premise is that guys like this might actually benefit from a good fight, and that bullying should be confronted by any means
The film's premise is that the three women spend a lot of time spying on others, but they are unable to see problems closer to home
You see how the film critics quoted have not simply described a detail of the situation of the film. They are saying something about the fundamental idea of the work, while communicating something deeper about what the experience of the story is like: tension and drama, a moral twist, an ironic contrast.
A premise can be very short, but if someone describes a premise in a way that sounds generic or bland, that will reflect their view of the work itself:
There is no dramatic tension to this film - the entire premise is already summarised in the title [Miracles From Heaven]
Some examples of what is missing
I've not seen the show, but I imagine a better premise might be:
- the struggle to survive when the dead return as zombies.
This communicates an idea of what drives the plot (characters motivated by survival, constant daily struggles) and what the experience of the show is like (tension, intensity, threat, some level of realism and grit), as well as the mechanics of the situation (the zombies part).
But maybe I've misread the publicity and the premise is actually:
- a grave digger has to dramatically change his business when the dead return as zombies, with hilarious consequences
- a couple's relationship is shaken when the dead return as zombies, and an old flame makes an unexpected return
- an underperforming sports team have a chance to change their fortune when the dead return as zombies
The premises of these shows are very different, but "the dead return as zombies" is true of all of them.