My 655 pound build (with comments):
http://uk.pcpartpicker.com/p/PwZst6
1) CPU:
The i5-4460 is, IMO, the best CPU at this price point. I can't recommend AMD at all since this CPU beats it in gaming in 99% of the games averaging 25% better depending on your games chosen which does vary a lot. It goes as high as 46% in Skyrim versus the FX-8350 (yes) and for a few games it's roughly the SAME such as BF4 (with the caveat of lower FPS for the minimum which means slightly more stutter).
2) Motherboard:
A few choices here, but there's a minimum I'll go on quality. This is it. Look at Customer Feedback etc, but I recommend Asus, Gigabyte or Asrock in general but carefully look at the quality as a dead board is bad news.
3) Graphics Card:
There's some CHEAPER ones you could go with definitely. I'd go with a GTX750Ti as the minimum. I could have gotten a cheaper R9-270X even but I wanted:
a) one with good quality, and good cooler
b) Factored in the GPU frequency (higher performance), and
c) really wanted at least 3GB of VRAM to future proof a bit.
Had the extra VRAM been too much I'd have dropped it. But all the above considered I like this card the best. There MAY be some NVidia specific options you want so that's worth considering (PhysX, Shadowplay, possibly G-Sync though a new monitor is required that's currently expensive)
So if NVidia, then a GTX750Ti or a GTX760 are my only recommendations (Asus for the either).
4) CASE:
There are cheaper, but this one is nice but also has two good FANS which I believe can be plugged into the motherboard for fan control. You really want TWO good fans which cheaper cases don't tend to have.
*Move the 12cm fan to the front as Intake. Plug both (and CPU) into the motherboard then setup fan control software.
5) Power Supply:
A couple choices but looking at one that's sufficient for the graphics card, decent reviews and reasonable price.
6) Memory:
8GB of DDR3 1600MHz CAS9 (2x4GB) is pretty standard for a good gaming PC. Better isn't needed arguably.
7) HDD:
1TB is pretty typical. You may want an SSD in the future but I didn't want to include one as I'd have to get a lower-end graphics card.
*Partition the HDD into two. I recommend 200GB for the first partition (Windows). I do recommend putting Steam on the 2nd partition though.
**Make a backup Image periodically of C-drive using Seagate DiscWizard (or Acronis True Image Free if WD drive). Put on 2nd partition. Ideally you want two drives but it does protect you in case of severe virus or corrupted software requiring a RESTORE.
8) CPU cooler:
The "EVO" I recommend is much better than the included stock CPU cooler. Both would work, but the stock one gets loud.
9) Windows 8.1 64-bit:
The ONLY Windows version I recommend. Much better than Windows 7 in many ways. I recommend Start8 (Stardock) to make the interface work like Windows 7 (Start Menu, disable Charms Bar... you can Google how to boot directly to your DESKTOP.)
SUMMARY:
This is my recommended build. There's a couple parts you might change but I strongly recommend the CPU for this budget. There are CHEAPER parts than I've recommended but I'm looking at a combination of quality and performance so take that as you will.
If you'd given me a smaller budget then it starts to get tricky. Slightly smaller and I'd mainly change the Graphics card to say a GTX750Ti. Really small and I'd recommend the G3258 CPU and the GTX750Ti likely (plus 4GB DDR3). The G3258 is a great budget gaming CPU (can Google more). For that, I'd get a suitable Asus H81 or similar because they can overclock that CPU slightly.
*I hope this gets you started. Again, if my build cost is too much at least now you know what I recommend as a minimum.