@madscouser69 posted above. It explains which parts it recommends, and why, to build your own budget PC.
The article recommends a Radeon RX 580 8GB for £183, but I would personally swap that for a NVidia GTX 1060 6GB for about £200.
They're targeting a price of around £600, give or take, so it's not a huge increase over your original budget, however you need to remember there's no monitor or mouse/keyboard included.
Also you'll need to factor in the cost of your operating system; if you're serious about gaming then it's got to be Windows 10.
If you decided to buy a prebuilt machine, instead of building your own, you will need to expect to pay more and might find their specs will be slightly different (lower) as they'll be trying to squeeze as much margin as they can into the build.