I've stumbled upon Goldman's paper Determinants in projective modules which defines $\det(\alpha)$ for all $\alpha\in End(P)$ and all finitely generated projective $k$-modules $P$. ($k$ is a commutative ring here)
Consequently one can define the characteristic polynomial of $\alpha$. The not completely obvious part of Goldman's paper is the insight that $\chi_0$ is in fact not of the form $X^r$ for $r=rk(P)$ as one would naively suspect but instead $\chi_0 = \sum_{i=0}^n e_i X^i$ for where $1=\sum_{i=0}^n e_i$ is a decomposition into pairwise orthogonal idempotents. Furthermore these are characterised by $\forall \mathfrak{p}\in Spec(k): e_i \notin \mathfrak{p} \iff \dim_{k_\mathfrak{p}} P_\mathfrak{p} = i$.
One can prove that an arbitrary characteristic polynomial $\chi_\alpha = \sum_i a_i X^i$ then satisfies $a_i \in e_i + \sum_{i<j\leq n} e_j k$. In particular they all satisfy $\deg(\chi_\alpha)=\deg(\chi_0)$ and the leading coefficient of $\chi_\alpha$ is equal to that of $\chi_0$.
A question that isn't answered by Goldman's paper, that I ask myself (and now you) is the converse:
If $P$ and a tuple $(a_i)_{i=0}^n$ of elements of $k$ are given s.t. $a_i \in e_i + \sum_{i<j\leq n} e_j k$, is there an $\alpha\in End(P)$ such that $\chi_\alpha = \sum_{i=0}^n a_i X^i$ ?
I can prove the weaker statement where $P$ is not fixed, i.e. given only the $e_i$ and the $a_i$, I can find a f.g. projective $P$ and an $\alpha\in End(P)$ such that $\chi_{0_P} = \sum_{i=0}^n e_i X^i$ and $\chi_\alpha = \sum_{i=0}^n a_i X^i$.
But since $\chi_0$ does not determine $P$ in general, this is not sufficient to answer the stronger question.