A good 6300 based computer is not gonna happen at your budget.Try this you could always upgrade to a 6300 later .
PCPartPicker part list:
http://uk.pcpartpicker.com/p/OKAv
Price breakdown by merchant: http://uk.pcpartpicker.com/p/OKAv/by_merchant/
Benchmarks: http://uk.pcpartpicker.com/p/OKAv/benchmarks/
CPU: AMD Phenom II X4 965 Black 3.4GHz Quad-Core Processor (£69.99 @ Maplin Electronics)
Motherboard: ASRock 970 PRO3 ATX AM3+ Motherboard (£51.56 @ Amazon UK)
Memory: Corsair XMS3 8GB (2 x 4GB) DDR3-1600 Memory (£32.99 @ Amazon UK)
Storage: Seagate Barracuda 250GB 3.5" 7200RPM Internal Hard Drive (£38.81 @ CCL Computers)
Video Card: Gigabyte Radeon HD 7790 1GB Video Card (£114.50 @ Ebuyer)
Case: Xigmatek ASGARD PRO (Black) ATX Mid Tower Case (£34.96 @ Dabs)
Power Supply: XFX 450W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply (£34.99 @ Maplin Electronics)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£11.77 @ Amazon UK)
Total: £389.57
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-04-06 13:11 BST+0100)