You have to bear in mind the processor you want, so start there. That processor will have a socket type (i.e. it determines what motherboards can take it).
Leading on from that if you look at motherboard specs for motherboards of that socket type it will tell you what type of RAM it supports and how much (i.e. how many slots there are)
Graphics cards generally work on most motherboards, it's the case that you need to watch out for as the higher end GFX cards require 2 slots and can be lengthy too.
You need a PSU with enough wattage to run your PC so these days about 500W for a gaming system.
You could steal your existing HDD.
If you look at Ebuyer's components menu it has sections for each major component and you can sort by price so just looking at the specs of the components you can figure it out gradually in your head by adding things to the cart.
If you build (i.e. add to your cart) a machine and printscreen the parts you used I will tell you if they are OK.
With your price range you could get a socket AM2 dual core AMD processor as they are cheaper at the moment and still play most games (i.e. not crysis). The 5000+ X2 i have in my machine atm cost me £150 2 years ago and is £42 now. I have one of those supporting an 8800 GTS and 2 gigs of RAM and it plays games wonderfully. Talking CoD4 in 1920x1200 full settings etc. You could get a similar setup with a cheaper but just as good Graphics card for cheap now I reckon.
You can build one then I will have a go after my exam tomorrow and we'll see what we can rustle up for your money. I'd do one now but sleep. Sleeep
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Stats to look for on a new PC