There is an algebraic topology book that specializes particularly in homology theory-namely, James Vick's Homology Theory:An Introduction To Algebraic Topology. It does a pretty good job of presenting singular homology theory from an abstract,modern point of view, but with plenty of pictures. It's an underrated book for this purpose that I wish would get more attention-you may find it very useful.
Another book I think you'll find very useful is the aforementioned Joseph Rotman's An Introduction To Homological Algebra. Not only is it the most accessible and clearly written book on the subject, it emphasizes the topological origin of many of the central concepts.
I think both of those,in addition to the other good suggestions here,will help you out a lot.