I must admit, I was in the same position last year, and ended up spending £1k on a machine that future proofed me. I went for an i5 processor (theres not much difference between an old 15 2500k, a i5 3570k and the new i5’s.
So you could have an Intel chip a couple of generations old and its still going to be fine. Dont bother with i7’s for gaming, the additonal hyperthreading does nothing for just about any game written today.
I bought a machine bundle that also had an Nvidia gtx670 graphics card fitted, and apart from the fact its been surpassed by a cheaper, but practicably no faster 700 series card, it still races through anything I play at full speed on a 1920c1080 monitor. Its over a year old and still realistically as good as you need for gaming.
Not that much help as your budget is a couple of hundred lower, but you should be able to get a machine to those sort of specs now.
Try Chillblast, Yoyotech, Overclockers as system builders, rather than lumbering into PC world, you’ll get a far better machine for the money.