Shop around all the computer stores, online or actual stores. I'm not really that familiar with UK options for those types of places, and even if I was I know that the currency exchange rate isn't the only factor and the parts will cost different amounts there. I'm going to explain the basics of what you need for a decent gaming computer and you can do the actual shopping.
First and foremost read reviews. This will give you an idea of how reliable the computer is before you buy it and have it break the day the warranty is up or have it break in a way the warranty conveniently doesn't cover.
You can buy a computer that's built already, or I'm sure there are websites that will let you customize a computer and have it built, or you can purchase the components and put them together yourself if you feel up for it. It isn't as hard as you probably think it is.
Anyhow you choose to do it, for full HD gaming you need a card with at least 1GB of video memory. The video card is the key component that will determine how your games run. From ATI, get something from 5600 series or 6600 series or higher. The first number doesn't say much about the power of the card, it's just the generation. 6450 is not better and faster than 5870, for example. Get the computer with the card that is the best you can afford. If you have a choice between a weaker processor with a better card or a better processor with a weaker card, go for the better card.
You only need a processor that is good enough not to bottleneck the video card. Pretty much any quad core from AMD or Intel i5/i7 will do just fine, even older processors like Athlon II and Core2Quad. If the quad cores are all too expensive, an i3 with hyperthreading or a phenom II X2 are both ok and both can be replaced with better processors later if you want.
The memory is the last thing you should worry about really. You need 4GB for lots of games but if there isn't enough it's the easiest and least expensive thing to upgrade on your computer. DO get a 64-bit operating system though, otherwise you max out at around 3.5GB of RAM.
All the other stuff is up to you - the hard drive(s), the case, the optional cooling systems, the optical drive(s), etc. Computers do NOT usually come overclocked and I don't recommend overclocking processors in gaming computers anyway since the games don't usually use the whole processor anyway. Anyhow if you decide you want to overclock make sure you get faster RAM and a good cooling system and read up about all the multipliers and frequencies and voltages and possible risks and max voltages and stuff before you do it.