Easily done as a starter build that will play most games.
500 or 600w 80+ psu usually £40 to £49 for instance the coolermaster b600 currently goes for £42
8GB (2 lots of 4gb) 1333mhz value corsair RAM £30 to £35
HD 7770 1gb GDDR5 £87
coolermaster K380 chassis £38 (only has one fan but space for 4 which means you can go for silent running by installing your own, better quieter fans as and when money permits
Coolermaster 212 evo cooler £20 to £25
z77 mobo £60
and cpu I3 £70 link below
http://www.overclockers.co.uk/showproduct.php?prodid=BG-071-IN&groupid=595&catid=686
There you go will do the job and is less than £350
Or if you prefer amd you can often pick up an fx4100 which has huge overclocking potential for £40 to £50
and could get asrock 990fx extreme3 mobo for £80. It has xfast utilities which I like but it must be pointed out asrocks customer services are a joke. This mobo is great for overclocking and can also add another card when money permits and upgrade cpu as well.
Many will have a problem with the ram but it does the job and is cheap. Ram is the easiest thing in the world to upgrade to in the future.
In fact my selections are all upgradeable but in the meantime you will be able to play all games at a good level, which is what a cheap gaming computer should do.
The likes of warcraft can be played in ultra easily with the above specs. Your fps will drop in 25 man raids ofc but you can always reduce settings when in those scenarios.
I haven't listed hard drives because they are so so cheap at the moment it is nearly impossible not to find a bargain or two.
Just make sure you get 7200rpm and not 5200. Strangely they are around the same price.
Also I would go for two lots of smaller than one bigger, for instance a 160gb for o's and 320 for games. But hey, you pays ya money you takes ya choice.
You should be able to get both for less than £40 in total. If this goes above the £350 (it wont by much), in fact I picked up a 320gb and 160gb just last week which cost £28 in total, but if it does and £350 is the max spend you can get a 970fx mobo for £56
That will bring you back to budget.
The only main difference is that the 990's pcie slots will give the same performance to two gpu's whereas the 970 is times 4 on the second one.
If you will only ever put one graphics card in then go for the 970 and you will be way under budget.