Here's one with an i5 4570:
PCPartPicker part list:
http://uk.pcpartpicker.com/p/3aOdN
Price breakdown by merchant: http://uk.pcpartpicker.com/p/3aOdN/by_merchant/
Benchmarks: http://uk.pcpartpicker.com/p/3aOdN/benchmarks/
CPU: Intel Core i5-4570 3.2GHz Quad-Core Processor (£137.99 @ Aria PC)
Motherboard: ASRock H87 Pro4 ATX LGA1150 Motherboard (£54.37 @ Amazon UK)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£58.96 @ Amazon UK)
Storage: Mushkin Chronos 240GB 2.5" Solid State Disk (£95.99 @ Aria PC)
Video Card: Gigabyte Radeon R9 270 2GB Video Card (£128.75 @ Ebuyer)
Total: £476.06
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-16 15:24 GMT+0000)
And here's one with what you were looking at:
PCPartPicker part list:
http://uk.pcpartpicker.com/p/3aOfR
Price breakdown by merchant: http://uk.pcpartpicker.com/p/3aOfR/by_merchant/
Benchmarks: http://uk.pcpartpicker.com/p/3aOfR/benchmarks/
CPU: AMD FX-6300 3.5GHz 6-Core Processor (£77.99 @ Aria PC)
Motherboard: Gigabyte GA-970A-DS3P ATX AM3+ Motherboard (£47.97 @ Amazon UK)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£58.96 @ Amazon UK)
Storage: Mushkin Chronos 240GB 2.5" Solid State Disk (£95.99 @ Aria PC)
Video Card: Gigabyte Radeon R9 270 2GB Video Card (£128.75 @ Ebuyer)
Total: £409.66
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-16 15:25 GMT+0000)
The FX 6300 does bottleneck in CPU intensive games. I know having owned one myself and from various reports from many other users. In some titles that don't strain the CPU as much it should be good. However you'd need to heavily overclock it to get it anywhere near the performance of the 4570.
As you've already pointed out, you could drop some extra cash into a better GPU. I personally would not be happy with the 6300, and I guess it's dependent on what you're going to play. Any particular titles? I would change the motherboard to a GA-970A-UD3 personally if you want to try and alleviate the bottleneck through overclocking (8+2 power phase vs the 4+1 in the DS3P), but you'd need a strong aftermarket cooler too.
To try and match the intel through overclocking you'd be looking at something like this, with a higher heat output, more strain on components (shorter lifetime) and more power draw:
PCPartPicker part list:
http://uk.pcpartpicker.com/p/3aOlp
Price breakdown by merchant: http://uk.pcpartpicker.com/p/3aOlp/by_merchant/
Benchmarks: http://uk.pcpartpicker.com/p/3aOlp/benchmarks/
CPU: AMD FX-6300 3.5GHz 6-Core Processor (£77.99 @ Aria PC)
CPU Cooler: Noctua NH-D14 65.0 CFM CPU Cooler (£57.89 @ Ebuyer)
Motherboard: Gigabyte GA-970A-UD3 ATX AM3+ Motherboard (£71.99 @ Maplin Electronics)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£58.96 @ Amazon UK)
Storage: Mushkin Chronos 240GB 2.5" Solid State Disk (£95.99 @ Aria PC)
Video Card: Gigabyte Radeon R9 270 2GB Video Card (£128.75 @ Ebuyer)
Total: £491.57
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-03-16 15:28 GMT+0000)
In the UK the prices are quite skewed, that it makes almost no sense to make an FX build unless you're on a very strict budget. Personally, I'd drop to a 120GB SSD and get a stronger GPU like a GTX 760 if you can squeeze it in, and go with the intel build. The fact is that the stronger single core performance of the intels is still extremely beneficial.