Here's a good build for almost exactly $600. It will be able to play most games at 1080p on high to medium-high settings:
PCPartPicker part list:
http://pcpartpicker.com/p/318ZN
Price breakdown by merchant: http://pcpartpicker.com/p/318ZN/by_merchant/
Benchmarks: http://pcpartpicker.com/p/318ZN/benchmarks/
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($118.49 @ Amazon)
Motherboard: Asus M5A97 LE R2.0 ATX AM3+ Motherboard ($74.97 @ OutletPC)
Memory: A-Data XPG V1.0 4GB (1 x 4GB) DDR3-1600 Memory ($32.99 @ Newegg)
Storage: Seagate Barracuda ES 1TB 3.5" 7200RPM Internal Hard Drive ($55.00 @ Amazon)
Video Card: MSI GeForce GTX 650 Ti Boost 2GB Video Card ($190.98 @ SuperBiiz)
Case: Cooler Master Elite 431 Plus (Black) ATX Mid Tower Case ($53.49 @ Staples)
Power Supply: Corsair Builder 600W 80+ Bronze Certified ATX Power Supply ($54.99 @ Microcenter)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($19.98 @ OutletPC)
Total: $600.89
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-02-27 15:22 EST-0500)