Every complex manifold that admits one Kahler metric $w$ admits a lot of them, indeed $w+i\partial \bar \partial f$ is a Kahler metric if the second derivatives of $f$ are not too large. This is why, asking if one can classify Kahler metrics on $\mathbb CP^n$ is more-less equivalent to ask if one can classify functions on $\mathbb CP^n$. Can we classify functions? It depends on what you want to know.
Even if we want to classify Kahler metrics on $\mathbb C^n$, what can this mean? One analogy can be helpful here. Namely this question is somewhat similar to asking if we can classify convex functions on $\mathbb R^n$. Such a function $f$ always define a Hessian metric on $\mathbb R^n$ given by $g_{ij}=\frac{\partial^2 f}{\partial x_i \partial x_j}$. So, can we classify convex functions?