The short answer is that although there are some polynomial time algorithms for provably finding approximate Nash equilibria, they all find relatively poor approximations -- probably not good enough if you are actually trying to find an algorithm to play a game. More is known for 2 player games than for n player games.
If what you are trying to do is actually find an (approximate) Nash equilibrium, one easy to code thing you might try is simulating game play, with each player using the randomized weighted majority algorithm (http://en.wikipedia.org/wiki/Randomized_weighted_majority_algorithm). This isn't guaranteed to work, but in many cases will (And is guaranteed to in certain classes of games, like zero-sum games). In particular, if this process converges at all, it is guaranteed to converge to a Nash equilibrium. The danger is that it will not converge, and cycle forever -- but even in this case, the empirical history of game play will converge to the set of coarse correlated equilibria.