I'm afraid for the money you're looking at, you're just not going to get an 'ideal gaming computer' for that price. You need to put more money on it if you want it to have any half-decent gaming capability.
When I buy a bunch of components to assemble a gaming machine (and I don't spend unnecessary money on excessively expensive components), I'm looking at putting between £900 and £1200 on it.
Now if you assemble it yourself and cut a few corners (like getting the cheapest case you can find), you could assemble a genuinely decent and capable gaming system for £520. You could push that down as low as £440 if you get the absolute bare minimum setup to game on.
Here's a bunch of components from one of my favourite online hardware retailers:
From scan.co.uk:
£103-178 GTX650/660 graphics
£87 i3 2120 CPU
£65 motherboard (http://www.scan.co.uk/products/msi-z77a-g41-intel-z77-s-1155-ddr3-sata-iii-6gb-s-raid-pcie-30-(x16)-d-sub-hdmi-atx)
£16 4GB RAM
£43 500GB hard disk
£0-52 120GB SSD (http://www.scan.co.uk/products/120gb-ocz-solid-3-ssd-25-sata-6gb-s-sandforce-2281-read-550mb-s-write-530mb-s-34k-iops)
£12 DVD
£12 case
£30-37 PSU (http://www.scan.co.uk/products/450w-antec-basiq-vp450p-gb-80-plus-eps-12v-quiet-fan-atx-v23-psu or http://www.scan.co.uk/products/430w-corsair-builder-series-cmpsu-430cxv2uk-80-eff-uncertified-quiet-fan-atx-v23-psu)
£73 Windows 7 64-bit OEM
£441-575 total
There are two main areas to adjust the price. Firstly the graphics card. Don't be fooled by the model numbers - GTX660 is a hugely more capable card and will deliver double the gaming horsepower of the GTX650. The GTX650 will let you play modern games, but you won't be able to play on max quality settings and it'll be obsolete in a couple of years. For now though, it's a capable enough card.
The GTX660 is a properly powerful card and will let you crank the graphics quality right up. It should serve you well for comfortably three years.
Second area to adjust price is the SSD - this is a storage device like a hard disk for installing Windows, software and games on. It's much faster and more reliable than a hard disk. It won't do anything to make gameplay faster or more responsive, but games will load much faster (as will your saved games and new areas you enter), Windows will boot much faster and software (like your web browser, Word, Photoshop, etc) will load and run much faster.
It's a pretty cool performance component you'd never normally see in a low-cost machine, but SSDs are getting a lot cheaper lately it's an amazing deal. You could safely leave the SSD out though if you really want to minimise cost.
Finally, I've put two options of power supply (PSU) in there simply due to stock levels at Scan. You could get similar components for a similar price from Aria, eBuyer, Novatech, Dabs, etc or even buy different components from different places for the best prices. Be aware of course you'd then be paying more than one company for delivery.
I know it's not what you were wanting to do, but think about it - if you could get a genuinely decent gaming machine for £300, why would anyone ever spend more!
If it's too rich for you and if you buy your games, then consoles are for your exact situation. For £300 you could get an Xbox 360 (£150 new) plus the Halo remake, Halo 2, Halo 3, Halo ODST, Halo Reach, Halo 4, all three Gears of War and still have money left over. PC gaming is even better but the initial cost is higher.
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