I'm assuming that you need to pick the appropriate $\alpha$ and $\beta$ yourself. Then here is how you can do it:
Define $\alpha$ to be the natural inclusion: $\alpha(n)=n$ for every $n \in N$. Define $\beta$ by saying that $\beta(g) g^{-1} \in N$ (for every $g\in G$ there is exactly one such $\beta(g) \in H$). All you need to do now is check that $\alpha$ and $\beta$ are homomorphisms and that the sequence is exact. You know what $\alpha$ and $\beta$ are, so this is a pretty straightforward task.
If you already know this, then I've told you nothing new. If that's the case then it would really help to see what you've tried yourself.