This fits your budget.... barely lol. But its all quality parts.
AMD 6 core processor, AMD 7950, Corsair power supply, it's also got cas 8 1600mh ram.
The lowest quality part of this build is the case, which is still a good case.
This is what I would buy with my money on this budget. If you can get a little more, then I would consider getting an SSD as a windows drive. Really speeds things up.
This will play battlefield 4 on high settings and several other games on high/ultra settings also
If you already have a monitor and a copy of windows, then you could get something like this for the same budget.
http://pcpartpicker.com/p/1Ddeo
This has a 8 core processor, 8 gigs of ram, a 7950 graphics card, and a 120gb ssd as a windows drive.