= Solution
Put $\Gamma=\pi_1(X,x)$ and $R=\mathbb Z[\Gamma]$. Give $\mathbb Z$ its trivial left $R$-<module> structure, through the augmentation $\varepsilon:R\to\mathbb Z$, $\sum n_g g\mapsto\sum n_g$. The <universal cover> $\widetilde X$ has the lifted cell structure, with $\Gamma$ acting by <deck transformations>.
For each cell of $X$, choose one lifted cell and orient every translate compatibly with its projection. The translates form one copy of the regular $R$-<module>. Thus
$$
C_k(\widetilde X)\cong\bigoplus_{\text{$k$-cells of }X}R
$$
is a free left $R$-<module>, and its boundary is $R$-linear. The augmentation $C_0(\widetilde X)\to\mathbb Z$ sends every vertex to one. The contractibility of $\widetilde X$ says that its reduced <homology (mathematics)> vanishes, so the augmented complex
$$
\cdots\longrightarrow C_2(\widetilde X)
\longrightarrow C_1(\widetilde X)
\longrightarrow C_0(\widetilde X)
\longrightarrow\mathbb Z\longrightarrow0
$$
is exact. It is therefore a <free resolution> of the trivial <module>. By the definition of the <Ext functor>,
$$
\operatorname{Ext}_R^i(\mathbb Z,\mathbb Z)
=H^i\!\left(\operatorname{Hom}_R(C_*(\widetilde X),\mathbb Z)\right).
$$
An $R$-linear cochain with values in the trivial <module> has the same value on every translate of a lifted cell. It is therefore exactly an ordinary integral cellular cochain on $X$. The correspondence does not depend on which lift was initially chosen. It also respects the coboundary: applying a constant-on-orbits cochain to the lifted cellular boundary sums precisely the incidence coefficients of the boundary in the base. Hence
$$
\operatorname{Hom}_R(C_*(\widetilde X),\mathbb Z)
\cong C_{\mathrm{cell}}^*(X;\mathbb Z)
$$
as <cochain complexes>, not just as graded groups. Cellular <cohomology> computes the <cohomology> of a cell complex, giving
$$
\boxed{H^i(X;\mathbb Z)\cong\operatorname{Ext}_R^i(\mathbb Z,\mathbb Z)
\quad\text{for all }i\geq0.}
$$
At $i=0$ both sides are $\mathbb Z$, by connectedness and $\operatorname{Hom}_R(\mathbb Z,\mathbb Z)=\mathbb Z$. This is the <cellular free resolution from a contractible universal cover> description of <group cohomology>.
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