Personally the terms "budget" and "gaming pc" shouldn't be used in the same sentence.
But if one must build one...
Recommended minimums -
CPU - Intel Core i3 from Microcenter - $99.99
Motherboard - MSI, Asus, ASRock or Gigabyte LGA1155 micro ATX ~$50-75
Video card - gtx 460 off of ebay - ~$100
- or - if you don't like used then go with the 7770 ~$125
Ram - major brand 4Gb DDR3 1333 or 1600 (2 sticks of 2gb each) ~$40
Hard drive - Western Digital Blue or SeaGate Barracuda 7200rpm 500Gb/1Tb ~$60-90
Power supply - major brand 500-550 watt ~$50
Optical drive - major brand DVD/CD burner combo ~$20
Case - major brand costing $50 or less
TOTAL COST - $469.99-549.99 without monitor, speakers, keyboard/mouse or operating system.
Including those items add approximately $200-250.
**If you are willing to spend a little more I'd look at the Core i5-3470 currently $149.99 at Microcenter and the 7850-7870 currently around $200-250.