@Tyler Lawson: I just saw this question. Our book published in 2011 and advertised on
does exactly that. No (or little) singular homology, no simplicial approximation. It gives many calculations of nonabelian second relative homotopy groups not available by traditional methods. It also gets to the Relative Hurewicz Theorem and the calculation of certain homotopy classes of maps, including the non simply connected case.
It is in a sense a rewrite of algebraic topology on the border between homotopy and homology, using functors defined in terms of homotopy classes of maps, and establishing their main properties directly.
Of course there is a lot of homotopy and homology theory it does not do, for example Poincare duality: I've put that as one of a number problems to solve in the style/techniques of the book!