The usual construction of cellular homology starts with the singular homology of a topological space, then uses relative homology groups $H_n(X^n,X^{n-1})$ for a CW-complex $X$ as the $n$-th chain group, and has to prove these cellular homology groups agree with the singular homology groups, as well as that the cellular chains are free abelian groups. If you want homology with operators from $\pi_1(X,x)$, then you have to choose the base point $x$ to consider the universal cover $\widetilde{X}_x$ of $X$ based at $x$, and the cellular homology of $\widetilde{X}_x$.

Here is an indication of "some other construction", given a full account in the book Nonabelian Algebraic Topology, EMS Tracts in Mathematics Vol 15 (2011). This construction is homotopically defined, so that invariance under homeomorphism is automatic; calculation relies on a Higher Homotopy Seifert-van Kampen Theorem, HHSvKT, whose proof uses methods of strict higher homotopy groupoids, and cubical methods. This theorem has nonabelian consequences in dimensions $1$ and $2$, not otherwise available, and in higher dimensions has as but one consequence the Relative Hurewicz Theorem.

The book has an extensive Introduction giving motivation and historical background, gives many calculations, relates to many classical results, and concludes with a number of problems and problem areas.

The idea is to start with a filtered space, $X_*$, i.e. a space $X$ and an increasing sequence of subspaces $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n\subseteq X_{n+1} \subseteq \cdots \subseteq X$. We then define $C=\Pi X_*$ to consist of the fundamental groupoid $C_1=\pi_1(X_1,X_0)$ in dimension $1$, and for $n \geqslant 2$ the family of relative homotopy groups $C_n(x)= \pi_n(X_n,X_{n-1},x)$ for all $x \in X_0$. These, with boundary maps and operations of $C_1$ on $C_n$ for $n \geqslant 2$ have the structure of what is called a crossed complex.

In a sense, this is a rewrite of the classical Poincaré approach to homology which uses "formal sums", i.e. free abelian groups, but using instead clearly intuitive notions of compositions of cubes to get the algebra. The idea of using $\Pi X_*$ as an algebraic structure goes back to A.L. Blakers, (1948), and to J.H.C. Whitehead, (1949), in his paper "Combinatorial Homotopy II", which develops ideas on homotopy classification, not so well known, even today.

The usual construction of cellular homology starts with the singular homology of a topological space, then uses relative homology groups $H_n(X^n,X^{n-1})$ for a CW-complex $X$ as the $n$-th chain group, and has to prove these cellular homology groups agree with the singular homology groups, as well as that the cellular chains are free abelian groups. If you want homology with operators from $\pi_1(X,x)$, then you have to choose the base point $x$ to consider the universal cover $\widetilde{X}_x$ of $X$ based at $x$, and the cellular homology of $\widetilde{X}_x$.

Here is an indication of "some other construction", given a full account in the book Nonabelian Algebraic Topology, EMS Tracts in Mathematics Vol 15 (2011). This construction is homotopically defined, so that invariance under homeomorphism is automatic; calculation relies on a Higher Homotopy Seifert-van Kampen Theorem, HHSvKT, whose proof uses methods of strict higher homotopy groupoids, and cubical methods. This theorem has nonabelian consequences in dimensions $1$ and $2$, not otherwise available, and in higher dimensions has as but one consequence the Relative Hurewicz Theorem.

The book has an extensive Introduction giving motivation and historical background, gives many calculations, relates to many classical results, and concludes with a number of problems and problem areas.

The idea is to start with a filtered space, $X_*$, i.e. a space $X$ and an increasing sequence of subspaces $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n\subseteq X_{n+1} \subseteq \cdots \subseteq X$. We then define $C=\Pi X_*$ to consist of the fundamental groupoid $C_1=\pi_1(X_1,X_0)$ in dimension $1$, and for $n \geqslant 2$ the family of relative homotopy groups $C_n(x)= \pi_n(X_n,X_{n-1},x)$ for all $x \in X_0$. These, with boundary maps and operations of $C_1$ on $C_n$ for $n \geqslant 2$ have the structure of what is called a crossed complex.

In a sense, this is a rewrite of the classical Poincaré approach to homology which uses "formal sums", i.e. free abelian groups, but using instead clearly intuitive notions of compositions of cubes to get the algebra. The idea of using $\Pi X_*$ as an algebraic structure goes back to A.L. Blakers, (1948), and to J.H.C. Whitehead, (1949), in his paper "Combinatorial Homotopy II", which develops ideas on homotopy classification, not so well known, even today.

May 20, 2015. For an Introduction to these ideas, starting with a section on "Anomalies in Algebraic Topology", see this presentation Galway, Dec. 2014.