Changed a few things, here's what I recommend you :
PCPartPicker part list:
http://pcpartpicker.com/p/2RMgh
Price breakdown by merchant: http://pcpartpicker.com/p/2RMgh/by_merchant/
Benchmarks: http://pcpartpicker.com/p/2RMgh/benchmarks/
CPU: AMD Athlon X4 750K 3.4GHz Quad-Core Processor ($79.98 @ OutletPC)
Motherboard: MSI FM2-A75MA-E35 Micro ATX FM2 Motherboard ($57.78 @ Newegg)
Memory: G.Skill Ripjaws X Series 4GB (2 x 2GB) DDR3-1600 Memory ($40.50 @ Newegg)
Storage: Seagate Barracuda 500GB 3.5" 7200RPM Internal Hard Drive ($47.00 @ Amazon)
Case: Rosewill FBM-01 MicroATX Mini Tower Case ($19.99 @ Newegg)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($14.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($84.98 @ OutletPC)
Total: $345.22
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-02-12 00:21 EST-0500)
Use your 6570 with it, and that should give you some... decent performance. Obviously, the budget is very small, don't expect too much. But at least, that won't bottleneck your 6570 at all.