When you have an explosion of objects (say objects whose count grows exponentially based on some parameter or parameters), this should point you in the direction that it's an NP-complete problem. When you have to inspect, check too many objects (combinatorial or others). Usually these objects are subsets or sub-spaces of some initial object space. You should build some intuition for this. But as usual, the intuition lies sometimes (I've been lied like this by my intuition on 2-3 occasions).
Then once you suspect some problem is NP-complete, just Google for it and try finding more information about the same or about a similar problem.
This is what I do at least and I've been solving quite a few algorithmic problems some time ago.
Here is a nice problem which I am pretty sure is NP-complete but which can be solved through a genetic algorithm for example.
http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=973
And as Dukeling said, there's no generic way of doing this.