Perhaps an example of why I might want this information would clarify the question.
I know that Hensel's Lemma allows one to lift to roots of a polynomial to roots in $\mathbb{Q}$ given a value for $p$ and an initial guess. My question is whether we can come up with a general rule that, say, limits the primes $p$ for which a rational polynomial can have roots in $\mathbb{Z}_p$. If I were to try, e.g., substituting the series
$$\sum_{k\geq 0} a_kp^k$$
into a polynomial expression, then is it a simple matter of applying formal power series arithmetic, combining like powers of $p$, then forming a system of equations where the resulting coefficients for each $p^k$ must cancel? Or am I missing some subtlety?
Your question is completely unclear.
Hensel lemma is saying that from $F\in \Bbb{Q}[x]$, that we transform in $f\in \Bbb{Z}[x]$ monic squarefree, there is some $N$ (easily computable but don't ask me its value) such that $f$ has a root in $\Bbb{Z}_p$ iff $f\bmod p^N$ has a root in $\Bbb{Z}/p^N\Bbb{Z}$. The correspondence is $a_N\to \lim_{n\to \infty } a_n$ where $a_{n+1}=a_n-f(a_n)/f'(a_n)$.