Just like the title says everyone throw out your best $250 dollar gaming rigs if you can....
A10 5800k APU with some garbage motherboard, 8 gigs of 1600, cheap ass case, 500 gig HDD, and 300w power supply (ALWAYS 80+). Overclock the RAM to at least 1866 for better graphic performance, and overclock the cpu to like 4.4 or 4.5.
Best I can do. That budget is sort of insane, honestly I would say look into something locally thats used for that price.
http://pcpartpicker.com/p/A1b9
Motherboard: $49.99 http://ncix.com/products/?sku=76851&vpn=FM2-A75MA-E35&manufacture=MSI%2FMicroStar&promoid=1270
APU: $99.99 http://ncix.com/products/?sku=76271&vpn=AD560KWOHJBOX&manufacture=AMD&promoid=1270
RAM: $49.99 http://ncix.com/products/?sku=62704&vpn=PXD38G1866ELK&manufacture=Patriot&promoid=1360
Case: $39.99 http://ncix.com/products/?sku=57910&vpn=Z9%20Plus&manufacture=ZALMAN%20TECH&promoid=1270
SSD: $65.99 http://ncix.com/products/?sku=78766&vpn=ASP900S3-64GM-C&manufacture=AData%20Technology
PSU: $39.99 http://ncix.com/products/?sku=30144&vpn=SS-350ET&manufacture=Seasonic%20Electronics
Total: $345.94
Sorry, that's as LOW as I'd be willing to go... That's pretty damn good if you ask me?
http://pcpartpicker.com/p/A1pE
Yeah not much you can get for that budget but would play some games.