You might be interested in [this paper of Jason Manning][1].  He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something like this) that there is an algorithm to construct the discrete faithful hyperbolic representation of $G$.

I don't think the strategy is at all what you suggest, though.  Instead, he builds the character variety

$\chi(G)=\mathrm{Hom}(G,SL_2(\mathbb{C}))//SL_2(\mathbb{C})$

and computes its decomposition into irreducibles.  Mostow Rigidity implies that a discrete faithful representation is contained in a 0-dimensional component.  This gives a finite list of representations to check.  Now, for each of these, his algorithm attempts to either construct a fundamental domain using hyperbolic geometry, or to find a proof that the representation is not faithful or discrete.

>So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.

This sounds *extremely* optimistic to me.  (Well, it depends what you mean by 'nice'.)

  [1]: http://arxiv.org/abs/math/0102154