Do you want to know the integral homology?
If you are happy with homology with coefficients in $\mathbb{F}_p$, the best way to compute (and describe) the homology of Eilenberg-MacLane spaces is the technique developed in a paper by Ravenel and Wilson (MathSciNet). See also Wilson's "sampler" (MathSciNet). They used the structure called Hopf ring (a collection of Hopf algebras equipped with other operation) to describe $\{H_*(K(\mathbb{Z},n);\mathbb{F}_p)\}_{n\ge 0}$ as a whole.
The point is they worked in homology instead of cohomology so that we can use
$$ K(\mathbb{Z},n)\times K(\mathbb{Z},m) \longrightarrow K(\mathbb{Z},m+n)$$
to "generate" $H_*(K(\mathbb{Z},n);\mathbb{F}_p)$ from $H_*(K(\mathbb{Z},m);\mathbb{F}_p)$ for $m<n$.
It turns out the Hopf ring structure is compatible with the Eilenberg-Moore spectral sequence and we obtain the Hopf algebra structure on the homology of Eilenberg-MacLane spaces easily. It is also important their technique works for generalized homology theories. In fact, their motivation was to compute the Morava $K$-theory of Eilenberg-MacLane spaces.
