Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Question

Let $A\in M_{n \times n}(\mathbb{C})$.
Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$.
Find a characteristic polynomial for $g(A)$.

I believe that the solution has something to do with Jordan forms, but haven't figured out how I should be approaching it.