It depends what needs to be included, in that £600. If that's just for the PC, itself, there are plenty of options. Obviously, if you are prepared to build it, yourself, you can get something better, both in performance, and suitability, for your needs, or you can get something reasonable, pre-built, from a specialist, like Chillblast or Dino. Main thing is, avoid the cowboys, offering cheap "gaming" PC's, on the likes of Ebay. Treat with great caution.
If your budget needs to include peripherals, like monitor, operating system, keyboard/mouse, then things become more difficult, and then you are almost forced into a self build, if you want decent performance.
If you confirm these requirements, can put together some "guidance"
This would be the best sort of build, from a reliable supplier, using an i5 processor, which has an advantage, over i3, for multi-player, in some games, like BF3.
However, apart from an advantage in some such game apps, i3 is just as good, and being cheaper, allows you to squeeze in better graphics. I will look for a decent i3 build, to give you the alternative.
Aaah. That will help a lot. To be honest, that one I did manage to locate, although, at least, using decent quality compononents, was fairly weak, but best I could find, al budget. Anything you could really call a gaming PC, of good quality, is well into te 700s, but without GPU, will be able to sort something decent at £600. Will get back to you, later today.
This is about as good as is possible, at the budget, without graphics card. It does, at least, include high quality case, motherboard, power supply, and processor. 4GB of RAM is adequate, 8GB would be better, but not possible, without "stretching" budget. I wouldn't want to "reduce" elsewhere, to accomodate that.
This gets the "core" right. You just need to add something like an MSI, Gigabyte, or XFX Radeon HD7870, or nVidia GTX660Ti, and you will have a good "gamer". Adding an SSD, would give it that extra "edge"