Defining cellular homology independent of singular homology Can cellular homology be defined without using simplicial or singular homology? One obstruction is to compute degree of attaching maps without invoking any homology theory.

This is done in the book partially titled Nonabelian Algebraic Topology, (EMS $2011$), for which some background is in  this paper. The main tool is a homotopically defined functor $\Pi: \text{Filtered Spaces} \to \text{Crossed Complexes} $ defined using fundamental groupoids $C_1= \pi_1(X_1,X_0)$ and relative homotopy groups $C_n =\pi_n(X_n, X_{n-1},X_0)$, and operations of the fundamental groupoids on the relative homotopy groups, and boundary maps  $C_n \to C_{n-1}, n \geqslant 2$ (due to Blakers, 1948, for the case $X_0$ is a singleton). Thus an element of $C_n$ is a certain homotopy class of maps.  In particular. when the filtered space $X_*$ is the skeletal filtration of a CW-complex, we get a strengthening of the usual cellular chain complex. 
To make calculations, one needs a Higher Homotopy Seifert-Van Kampen Theorem for $\Pi$, which is proved by cubical higher groupoid methods. So it is complicated, but the original two papers, (RB and P.J. Higgins, JPAA, 1981) take just 61 pages, and do more than the classical cellular theory, particularly in dimension 2. 
Of course the book includes much more than those papers, particularly tensor products, homotopies,  and relations with chain complexes with operators. 
