I dont this think gaming is possible under £500, let alone $500.
Well you could place some games on minimum, but certainly not any of the latest release like BF4 / COD / GTA / GT6 ect ect. Well not at a decent level of graphics / FPS and overall experience.
If I only had $500 and wanted to get into game, I would be very tempted to pick up a next gen Counsel such as the PS4.
You simply can't build a a machine that can match the performance of current / next gen machines for the same price as they cost, although of course you can build much better machines overall, but as you said, gonna cost £1500-2000 for a Rig capable of playing the latest games in 1080p at a high FPS in ultra / Max setting.
---------- Post added at 11:25 PM ---------- Previous post was at 11:23 PM ----------
And just so you know, I dont have the money for a nice gaming rig, but while I save up over the next copula years or whatever I will be getting a PS4 for now.