The CPUs of a computer's processing unit is the only major component that will not affect FPS directly.
The two most important components to look for with regard to graphics performance are memory and graphics cards. If your computer recognizes the maximum amount of memory your system can use, the one way it might be able to improve gaming is by upgrading its RAM or video card. In general, players with old computers should stick closer to games with low resolution and older-style art styles in order to have a smoother experience playing them. With new games being released almost monthly for less than $10 online, there has never been a better time to spend a weekend upgrading that GPU...