Right I've been busy, making an half decent machine that's £300 is difficult so I set myself a slightly higher limit as it would have to be used as an office machine which would save money using it for gaming and office work. These prices are for the base machine only and do not include a monitor or KB/Mouse.
Budget machine can be seen here : https://www.aria.co.uk/WishList/ypwcWXJfYGw4Pa_UGPojYw,,
It's a quad-core Trinity based machine so it offers a good mix of power and graphics performance, as a gaming machine it offers more power the the current consoles do and would have no problems at all playing consoles ports at higher then console settings.
It has a Blu-Ray drive for media functions and 8Gb of DDR3 that runs at 1866Mhz to provide a boost to GPU performance via increased memory bandwidth.
It also is micro ATX in form factor making it smaller then the average machine, has a PCIEX slot to add a dedicated GPU in the future.
Mid-range system can b seen here : https://www.aria.co.uk/WishList/LKDIJONlA067gq7ZdYIKfg,,
Phenom 2 x4 Black edition, much better gaming CPU then Bulldozer!
8Gb RAM, 2x HDD's in RAID 0
Blu-Ray drive and an HD 7770 which is a massive boost over the GPU in the budget PC
All in one water cooler for CPU over clocking
High end system can be seen here : https://www.aria.co.uk/WishList/IE3WpyWfgAXnH8Rb82luSw,,
The system is £932 with an HD 7870 GPU, this price will vary depending on GPU you ultimately want to go with, add another £40 for a HD 7950.
Comes with a Blu-ray drive and an SSD for blazing OS load times, plenty of storage for games.
Is water cooled with a Corsair H100 for CPU overclocking while being quiet, the system is still micro ATX and is still small.
Not only will this system blaze through any console port it will also rip any pc games at 1080p fully maxed out.
My 2 cents
I built these very very quickly, shopping around you could get them built cheaper, even though the budget system is over the £300 price set by Shifty it double up as a gaming machine
AND office PC, with a console a PC is still required and that in itself is a £300-400 out lay.
You can spend more then the high end like myself but moving to a higher end Intel CPU won't improve gaming all that much and a 7870/7950 would be more then enough for 1080p.
Shifty that budget machine would be ideal for you, enough grunt for some decent gaming and the capability of an office PC all in one small package.