For that price, I don't think compromises need to be made (depending on existing equipment/peripherals, but it sounds like that price might be excluding so I'm going to assume you've got a monitor and other peripherals and just build the gaming box).
You really want to start with a budget bare-bone box. Intel H55 chipset, 4GB DDR3 1333MHz, i5 quad core, 450+ Watts of PSU, and a box. It's a really easy way of buying a cheap case and getting a deal on the more budget orientated end of the component scale in the bundle. My Google-fu is failing me, I'm going to link a less parts bundle and point out it is using the old ASUS bundled PSU which is no good. 380W may well run this system in the end, but you really should be looking at a 450W sticker on the box to be safe, I'd say a 550W/600W is a good bet if you want it to also run your next rig (cases often last forever, as BTX died and we're still using the same design as a decade ago, one of my rigs runs in the case I got 10 years ago, and I try and get two generations of rigs out of each PSU*).
Prices are taken at time of posting, linked to a website I've used plenty of times but taht does not indicate they are either the best or cheapest (their selection isn't too great). Here is a full component purchase as I couldn't find a good starting barebones to build from:
P55 mobo (bit older than H55 stock, can still take the CPUs we want and runs fine) £77
£22 gets you a [link=http://www.aria.co.uk/Products/Components/Cases/Arianet/Arianet+1011B+Midi+Tower+Case+(500W+PSU)+-+Black?productId=39884">basic case with crappy 500W PSU, while £20 + Quad Core i5 760 £140 or £80 1GB GTX460 £175 or ATi dude?
#6436870, By Shivoa 4850 v GTX 460 (it's not as stupid as it sounds)
Shivoa 6,314 posts
Seen 5 days ago
Registered 15 years ago