Algebra for algebraic topology

If you need a concise but very clear book which covers a lot of Algebraic Topology and just the necessary algebra (spectral sequences as well) I think that Differential Forms in Algebraic Topology- Bott & Tu is the book you are looking for.

Edit: It seems that Bredon-Topology and Geometry is closer to that you are looking for.

I think what you need is a book on Homological algebra that discusses some category theory, some homology and group cohomology. You can try

A Course in Homological algebra by Peter Hilton and Urs Stammbach

You can read first two chapters (first chapter talks about modules second chapter talks category theory) and then read 6th chapter on group cohomology. These comes around 130 pages.

There is a 5 page section named "Homological Algebra and Algebraic Topology". This is about application of Homological algebra in Algebraic Topology. You can read this first.