Greetings to a fellow Scot
The most significant thing in your information is how small your monitor resolution is. This is both good and bad. What it means is that at some point in the not too distant future you are probably going to want to think about upgrading it, which is an extra expense. The good, or kinda good, side of this is how easy it is to max out games at such a low resolution. The most common resolution these days is 1080P, or 1920*1080, over 2.5 times the number of pixels you have to fill, so your graphics demands are pretty low.
Now, if you want to spend up to your £600 and more, it makes sense to get a monitor, or else you end up spending money you can't really use. Here's a suggestion for how that kind of build might look-
It has an i5, and a HD7870XT (which is really a HD7950-light) and offers pretty excellent value for money. The monitor is as cheap a 1080P as you can get really, but it should do the job.
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: Intel Core i5-4440 3.1GHz Quad-Core Processor (£133.53 @ Ebuyer)
Motherboard: ASRock H81M-HDS Micro ATX LGA1150 Motherboard (£37.30 @ CCL Computers)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£60.25 @ Amazon UK)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive (£42.26 @ CCL Computers)
Video Card: PowerColor Radeon HD 7870 XT 2GB Video Card (£140.95 @ CCL Computers)
Case: Cooler Master K280 ATX Mid Tower Case (£32.25 @ Amazon UK)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply (£48.99 @ Amazon UK)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£13.18 @ Scan.co.uk)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) (£67.93 @ CCL Computers)
Monitor: Hannspree HE225DPB 21.5" Monitor (£89.98 @ Amazon UK)
Total: £666.62
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-14 00:30 GMT+0000)
Alternatively, if you want to keep your current display, and just get games running well on it, spending anything like that kind of money is a bit pointless.
You could honestly cut this down to a sub £400 build, but it wouldn't give you much going forward, so here is a price conscious build which will play every game fine at 1024*768 but also gives you a bit of space for expansion. With that in mind, it has a bigger PSU than it needs, and probably a better motherboard too
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: AMD FX-6300 3.5GHz 6-Core Processor (£82.79 @ Aria PC)
Motherboard: Gigabyte GA-970A-DS3P ATX AM3+ Motherboard (£50.50 @ Amazon UK)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£60.25 @ Amazon UK)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive (£42.26 @ CCL Computers)
Video Card: Sapphire Radeon HD 7770 GHz Edition 1GB Video Card (£67.90 @ Amazon UK)
Case: Cooler Master K280 ATX Mid Tower Case (£32.25 @ Amazon UK)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply (£48.99 @ Amazon UK)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£13.18 @ Scan.co.uk)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) (£67.93 @ CCL Computers)
Total: £466.05
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-14 00:36 GMT+0000)