As long as $f$ is monic of degree $n$, and irreducible mod $p$, the $\mathbb{Z}/p^k$-algebra $(\mathbb{Z}/p^k)[x]/f(x)$ is flat and has perfect mod $p$ reduction $\mathbb{F}_{p^n}$. The theory of Witt vectors tells you that there is a unique such flat $\mathbb{Z}/p^k$-algebra, which can be identified with $W_{k}(\mathbb{F}_{p^n})=W(\mathbb{F}_{p^n})/p^k$. There is a Galois-theoretic way of thinking about $W(\mathbb{F}_{p^n})$: It is the unique unramified degree $n$-extension of the $p$-adic integers $\mathbb{Z}_p$.

The point is that the ring we get depends only on the mod $p$ reduction of your original polynomial in quite a canonical way.
As soon as you care about $\mathbb{Z}/p^k$-algebras which are not flat, or whose mod $p$ reduction is not perfect, this breaks down and things become messier.