I might be misunderstanding your question - are you asking about a (co)homology theory for commutative monoids, sorry]or about trying to do homological constructions in the category of commutative monoids?
In the former case, I think Grillet has some work on this, see e.g.
Grillet, Pierre-Antoine(1-TULN) Commutative semigroup cohomology. (English summary) Comm. Algebra 23 (1995), no. 10, 3573--3587.
The idea is basically what Mikael describes, although I only know the more primitive version which is cotriple/comonad cohomology as set up by Barr and Beck. Presumably the problem with trying to apply that set up is ensuring there are enough abelian group objects to use as coefficient modules.