Good luck. £300 wouldn't even get you a desktop that you could call a gaming machine never mind a laptop. A decent graphics card such as a GTX970 or 980 will cost you £300+ alone. If you spend 300 on a laptop expect for it be for word processing etc only. It may run older games but nothing exciting. You also have to remember you can't really upgrade a laptop either unless you go super high end.
My advice would be to save your money for a desktop and aim to spend £700+ for decent machine. You could look at finance from scan or overclockers and use your current cash as a deposit.
If you really want a laptop expect to spend over £1000 for anything decent.