Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological spaces.
But the usual definition of singular homology is on the category of topological spaces, and you can show that it is homotopy-invariant only after having defined it on the category of topological spaces.
For example, the definition uses the free abelian group on the underlying set of the space of singular n-simplexes, $n$-simplexes, and taking the underlying set of a space do not make sense in the homotopy category.
I would like to have a construction of singular homology that can be entirely carried out in the homotopy category.
I was thinking of categorifying the usual construction:
Take the free spectrum ($(\infty,1)$ equivalent of abelian group?) on the space of singular n-simplexes, show that this is an "$(\infty,1)$ chain complex" and compute its "$(\infty,1)$ homology".
But I have the impression it will not work as is, becauses because simplexes are only interesting as topological spaces, not as homotopy types.
Is there a way to construct singular homology (or in fact any homology theory) directely directly in the homotopy category without using the category of topological spaces?