Acyclic group and finite CW-complex

I presume by "acyclic" you are referring to homology with $\mathbb{Z}$ coefficients. There are many such examples.

For instance, you can take two elements $u,v$ in the free group $F_2$ of rank 2 that satisfy the $C'(1/6)$ small-cancellation condition, and also such that $u,v$ together generate the abelianisation $\mathbb{Z}^2$. Explicit examples are easy to construct.

The small-cancellation condition then implies that the corresponding presentation complex $X$ is aspherical, and the assumption about the abelianisation implies that $X$ has the homology of a point. But $\pi_1X$ is an infinite hyperbolic group, in particular non-trivial.

The Higman group with presentation $$\langle{a,b,c,d}\mid{aba^{-1}b^{-2}},~bcb^{-1}c^{-2},~cdc^{-1}d^{-2},~ dad^{-1}a^{-2}\rangle$$ is perfect, and the 2-complex associated to this presentation has Euler characteristic 0. Hence this complex is acyclic. It is in fact aspherical, but it may be simpler to observe that Higman's group is also an iterated generalized free product with amalgamations $(A*_{\langle{b}\rangle}B)*_{F(a,c)}(C*_{\langle{c}\rangle}D)$, where $A,B,C$ and $D$ are copies of the Baumslag-Solitar group $BS(1,2)$, generated by $\{a,b\}$, $\{b,c\}$, $\{c,d\}$ and $\{d,a\}$, respectively. We may assemble a 2-dimensional Eilenberg-Mac Lane complex for the Higman group in a similar way.