You can try with Brown's book
Chapter V, Section 6, p. 121 ("Application: calculation of the homology of an abelian group").
Maybe take also a look at the following papers by Baumslag, Dyer and Groves:
The integral homology of finitely presented metabelian groups I
Amer. Journal of Math. 104 (1982), 173-182
The integral homology of finitely presented metabelian groups II
Amer. Journal of Math. 109 (1987), 133-156
and at the references therin.
EDIT. J. Schafer's thesis
J. Schafer, On the homology ring of an abelian group, Dissertation, University of Chicago, Chicago, Ill., 1965
seems strictly related to what you are looking for. However, I could not find any published paper with this title. The only Shafer's paper related to homology of abelian groups seems to be
J. Schafer: Abelian groups with a vanishing homology group
Canad. J. Math. 21(1969), 406-409.
