A CM type for K is a choice of one out of every pair of complex conjugate embeddings of K. For each CM type on K, there is an abelian variety of that type. This is a complex abelian variety B of dimension g with an action of K (rather, its integers) such that K acts on the tangent space of B through the g chosen complex embeddings. Now you can take A to be a product of B's corresponding to whichever CM types you choose. That certainly gives you a lot of possibilities for the multiplicities.

Added: In fact, that gives all multiplicities subject to the obvious condition that the multiplicity of an embedding and its conjugate add to n/g. However, the questioner points out that he wants an abelian variety A such that $K=End(A)\otimes Q$. Take an A as constructed in the first paragraph. Then $V=H_1(A,Q)$ is a K-vector space with a Hodge structure and a compatible Riemann form, which we can use to define a Shimura variety (= moduli variety). The problem now is to prove that (as expected), the family of abelian varieties on this variety contains one satisfying the condition....