Recovering a group from its quotient group Suppose I've a group $G$ and a normal subgroup $H$. I know the structure of $G/H$ as well as the structure of $H$. Is it possible to recover the original structure of G from this?

There is one important piece of data that you have omitted: namely, if $H$ is normal in $G$, then $G$ acts on $H$ by conjugation, and so there is a homomorphism
$G/H \to Out(H)$ (the group of outer automorphisms of $H$); without remembering this, there would be no chance of reconstructing $G$.
Now in general giving $G/H$, $H$, and a map from $G/H$ to $Out(H)$, there is
still no chance of reconstructing $G$.  E.g. even if $G$ and $H$ are abelian,
so that the homomorphism $G/H \to Out(H)$ is trivial, Zev's answer gives
a counterexample.  
However, if $G/H$ and $H$ are finite of coprime order, then in fact any extension
of $G/H$ by $H$ is necessarily a semidirect product, and so in this case your question (suitably modified by remembering the conjugation action) has a positive answer.  This is the Schur--Zassenhaus theorem.  
This is why counterexamples (such as in Zev's answer) tend to focus on $p$-groups.
