Take any group $G$ (non-abelian if you like) that has a presentation $G = \langle g_1, \dots, g_s \ | \ r_1, \dots, r_s \rangle$ with the same number of generators and relations.
Form a CW-complex $X$ with one $0$-cell, $s$ 1-cells (representing the generators $g_i$) and $s$ $2$-cells that represent the relations. Then $\pi_(X) = G$ and the cellular chain complex that computes $H_{\ast}(X)$ looks as follows: $$\mathbb Z^s \xrightarrow{\partial} \mathbb Z^s \to \mathbb Z$$ The differential $\partial$ can of course be calculated easily by abelianizing the relations $r_i$. If $G$ is finite, then $H_1(X;\mathbb Q) = G^{\text{ab}} \otimes \mathbb Q = 0$, hence $\partial \otimes \mathbb Q$ is surjective and hence injective, hence also $\partial$ itself is injective, hence $H_2(X;\mathbb Z) = 0$. Thus, $X$ is as desired.
It remains to undertstand which (finite) groups $G$ admit such a presentation.