Very Tight budget .... what is the most advanced game you plan on playing? Are you interested in buying a used or refurb computer and adding a high end video card to that to try to squeeze it into the small budget and get the best CPU and GPU combination that your money can buy or are you wanting to build a system new only?
I toyed around with different builds and numbers I come up with are:
Windows 10 OS =$100
Gigabyte AM3+ Motherboard = $56
Crucial DDR3 8GB RAM 1600Mhz = $40
Computer Case with 500 Watt PSU = $66
1TB Hitachi Hard Drive 7200RPM = $45
ASUS DVD-RW Drive = $20
The necessities without CPU and GPU put it at $327.00 Now to get a good CPU and GPU in the remaining $193.00, you could get a Quadcore FX-4350 4.2Ghz CPU for $80 leaving you with $113 for a video card, but your going to struggle to find a new video card that fits into $113 that will give you the 40-50fps 1080p
This build I put together cuts as many corners as you can without making a crippled build. Saving money in avoiding Intel and going with AMD, getting a case that comes with a 500 watt power supply to cut off about $30 or more, and going with Hitachi 1TB drive vs Seagate or Western Digital to get you maximum capacity and performance for your money, and 8GB RAM is the bare minimum for a gaming build like this. You might be able to squeeze by on 4GB, but to avoid lots of paging so much 8GB is best to start from. Gigabyte motherboard is a brand and quality I trust and never had any problems with and this is their entry level gaming board. I also have this board with my FX-8350 8-core build and I like the board, its feature rich for the price tag and no problems.
Thinking your going to need to add about $130 more to your budget to try to get a Geforce GTX 970 video card to get the FPS you want at 1080p. The CPU FX-4350 is entry level gaming CPU at $80 and for $70 more you can get the 8-Core FX-8350 4.0Ghz. If planning on going with 8-core then you need $200 more in budget so $720. If you want 16GB RAM add $40 more and that puts it to $760 and removes some of the corner cutting gaming build. If you want more speed from there, swap put the HDD for a SSD and spend even more.
Here is list of parts I was using for price reference:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814500362&cm_re=970_gtx-_-14-500-362-_-Product Video card $237.00
http://www.newegg.com/Product/Product.aspx?Item=N82E16811156258 case with 500 watt power supply $66.00
http://www.newegg.com/Product/Product.aspx?Item=N82E16813128565 motherboard $56.00
http://www.newegg.com/Product/Product.aspx?Item=N82E16820148540 8GB RAM $40.00
http://www.newegg.com/Product/Product.aspx?Item=N82E16832416892&cm_re=windows_10_home-_-32-416-892-_-Product Windows 10 OS 64-bit $100
http://www.newegg.com/Product/Product.aspx?Item=9SIA5AD2KD3645 1TB Hitachi HDD $45
If interested in buying a used system and upgrading it to be a gaming box, you might be able to get a used earlier generation Intel Core i5 desktop, upgrade its power supply, RAM, and Video Card and get it to squeeze into $520 very tightly.
