So for a budget of between £250-300GBP (correct me if I'm wrong), your probably better off going for used. A couple of builds here which you should consider:
- http
/www.ebay.co.uk/itm/Intel-i3-Gaming-PC-Full-Package-8GB-RAM-3-7-GHZ-GEFORCE-Graphics-/252367055149?hash=item3ac23fa92d:g:7LMAAOSwJQdW~nu~
(This one is what would likely be ideal for you since it is a fairly budget system but should perform quite well in most games. It is a real bargain at this price especially if it is as new as advertised).
-http
/www.ebay.co.uk/itm/Used-Gaming-PC-/121961879269?hash=item1c657e96e5:g:7yYAAOSwBnVW~D0F
(Has a suprisingly expensive GPU for this price but doesn't say exactly what CPU it is, could be a good buy)
-
http/www.ebay.co.uk/itm/10-4GHz-Quad-Core-Gaming-PC-4GB-RAM-500GB-GT-Geforce-HDMI-/152030544641?hash=item2365ba0301:g:8K0AAOSwiYFXF176
(Bit of a wild card this one. It is only £115 which means that you could afford to add a new GPU and PSU for about ~£160 and still be close to what you wanted to spend originally. The Q8400 isn't the greatest CPU but might be alright).
The first one should be your ideal build since you get a fairly decent CPU and GPU. The other two make compromises on the CPU. Feel free to keep looking but here's just a few ideas to throw out there. By no means are these builds the best out there, but they may be just good enough for what you are looking for.