Exercise 2 §III.1 'Cohomology of Groups' K.S. Brown:
Let $G$ be a group and $M$ a $G$-module. Show that $H^1(G,M)$ is isomorph to the group of derivation from $G$ to $M$ modulo the subgroup of principal derivations (principal derivation means: m \in M be fix, $d:G \rightarrow M$: $d(g)=gm-m$).
You have the bar resolution $B_\bullet$ of $\mathbb Z$ as a $G$-module, which is a $G$-projective resolution. Use it to compute $H^1(G,M)$: apply the functor $\hom_G(\mathord-,M)$ to $B_\bullet$ and write down exactly what it means for a $1$-cochain in that complex to be a $1$-cocycle, and what are the $1$-coboundaries.
If you know what the bar resolution is, this is just a matter of chasing down definitions.