You might be interested in this paper of Jason Manning. 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'.)
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ADDED MUCH LATER
I just learned that another algorithm for finding hyperbolic structures is given in this paper of Luo, Tillmann and Yang.