Here's what PC advisor magazine rates as the best gaming computers. (The article is only a fortnight old):
http://www.pcadvisor.co.uk/test-centre/desktop-pc/best-gaming-pcs-in-uk-2015-uk-may-3338847/
The only two within your budget are numbers 10 and 1 in the list. Neither of them include peripherals but, if you've already got a decent monitor, that probably won't matter.
The Amazon price of £374.95 for No.10 doesn't include an operating system but adding on Windows still won't take you anywhere near to the upper end of your budget. The price of £671.00 for No.1 also doesn't include Windows but there's an option to add on Windows for a total price of £754.00.
I'll leave it to others, who will be far more knowledgeable than me about gaming computers, to tell you whether it's worth paying the extra 300 quid for the more expensive model.