This is not a classification, but you can get a grip on the space of Kahler metrics on $CP^N$ using Bergman metrics and the Segre embeddings.

To explain this conside let $\{s_\alpha\}$ be a basis of homogeneous polynomials of degree k in N+1 variables. This gives an embedding $CP^N\to CP(H^0(O(k)))$. Now define a metric on H^0(O(k)) by declaring that the s_i are orthonormal. This defines a Fubini-Study metric on P(H^0(O(k)) and which pulls back to a metric on $\omega'$ on $CP^N$.

Now it can be proved that any Kahler metric $\omega$ on $CP^N$ is the limit of metrics of the form $k^{-1}\omega'$ for suitable bases $\{s_\alpha\}$ as $k$ tends to infinity (You can take the $\{s_\alpha\}$ to be orthonormal with respect to the $L^2$-metric induced by $\omega$).

In fact there is nothing special about $CP^N$ here. The above works for any projective manifold $X$ with ample line bundle $L$.