You can build them using a homology decomposition and read off the number of cells in each dimension from the homology groups.
For any simply-connected space, this will give you a construction by iterated cofiber sequences $M_n \to X(n-1) \to X(n)$ where $M_n$ is a Moore space and $X = \mathrm{colim} X(n)$. If you construct $M_n$ efficiently and give $X(n)$ the inherited CW structure, then this will be the absolute fewest cells possible in each dimension to construct a space with the required homology.
By an efficient construction of $M = M(G,n)$ where $G$ can be generated by $k$ elements but not fewer, and an exact sequence $0\to F_1\to F_0\to G\to 0$ where $F_0$ is free of rank $k$. Then we topologize this to get a cofiber sequence $W_1 \to W_0 \to M$, where $W_0$ is a wedge of $k$ $n$-spheres and $W_1$ is also a wedge of $n$-spheres.