I don't really know what makes a game run better, so I really need help here.
You could look at my rig. Under $800 CDN.
AMD Phenom II BE x4 3.2 Ghz 6Mb Cache (can be overclocked to 4.1ghz using AMD drivers) ($125)
XFX Radeon HD 6870 1Gb ($150) - for a little more you can get the 2gb version
Kingston 8Gb 1600Mhz ($60)
Corsair 600w ($80)
Seagate Barracuda 1TB 7200RPM ($80)
Cooler Master Hyper 212 ($40)
MSI 870-G45 (gaming Mobo) ($50)
The one from Tom's article is much better. Though, few days ago there were some new parts that came out, so I'd suggest a few corrections. It depends on your budget of course! This one is $70 more expensive, but on top of that I added windows as well, so the price increased a little bit more:
No get the build Sunius suggested its perfect for that budget. What makes a good gaming rig is first the graphics card (more RAM and higher speed is not always better) second the CPU but don't compromise on the graphics card. RAM 4Gb is good enough but its cheap so 8Gb will future proof it any more is overkill. Better hard drives increase loading times but don't improve gaming. Read this article for info on good graphics cards http://www.tomshardware.co.uk/gaming-graphics-card-revi...
As a general rule you want to spend around the same amount on the graphics card as your CPU and motherboard combined. Also avoid i7s for gaming as they are no better than i5s and cost alot more.