Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or permutation representation, or more generally as the automorphism group of some object. (Obviously the problem of showing two different presentations are equivalent is well-studied already and I know where to look for that.)
Of course there's no general technique here that's of any use. I'm just interested in seeing nice proofs in particular cases, to get an idea of the different ways it can be done.
Perhaps this isn't what you're looking for, but the question is pretty broad, so here are some general tips. When I know a presentation is supposed to describe a finite group, the first thing I usually do is try to figure out $[a,b]$ for all $a,b$ in the generating set. Then I try to find the generator orders, and from there the order of the group. Once I know that, I'll start looking for the center, derived subgroup, sylow subgroups, etc. Hopefully by that point the group will be recognizable. Sometimes using Smith normal form can help simplify the relations. For an example of this type of reasoning you might be interested in reading this answer of mine.