500-600£(UK) pc build

Yes it is possible, i would recommend first buying a Asus gt 750ti graphics card, it will give you your desired FPS but just be careful when buying. Keep to good quality and well known brands but if something is too expensive do not be afraid to wait for it to drop the price.

Let me know your thoughts.

Yes, It's possible.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor (£49.99 @ Amazon UK)
Motherboard: MSI B85-G41 PC Mate ATX LGA1150 Motherboard (£54.67 @ CCL Computers)
Memory: Kingston HyperX Fury Black 8GB (2 x 4GB) DDR3-1866 Memory (£35.69 @ Amazon UK)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£39.98 @ Scan.co.uk)
Video Card: Gigabyte GeForce GTX 950 2GB OC Edition Video Card (£125.99 @ Aria PC)
Case: Cooler Master N400 ATX Mid Tower Case (£34.99 @ Scan.co.uk)
Power Supply: Super Flower Golden Green HX 450W 80+ Gold Certified ATX Power Supply (£47.71 @ CCL Computers)
Monitor: LG 24GM77 144Hz 24.0" Monitor (£205.17 @ PC World Business)
Total: £594.19
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-21 17:19 BST+0100

This build isn't as good as RCFProd's, but it'll do what you want for a little cheaper, to the point at which if Linux isn't you're cup of tea and you're not a fan of pirating you could still buy windows and still be close to within budget. Or you can save the extra money for something else. Or use it to upgrade your cpu/mobo combo and your gpu to the something like the second build listed below, which is on par with RCFPRod's build, if not a little better for the same price. Up to you, granted both builds only have 4gb of ram, but even the recommended specs for csgo say you only need 2gb of ram. So if you really do just want to play games like csgo at 1080p at 120fps, either of these builds will do it, and give you the option of using whatever OS you choose, along with that lovely monitor you wanted.

PCPartPicker part list: http://uk.pcpartpicker.com/p/j6WjrH
Price breakdown by merchant: http://uk.pcpartpicker.com/p/j6WjrH/by_merchant/

CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor (£53.99 @ Amazon UK)
Motherboard: Gigabyte GA-F2A88XM-D3H Micro ATX FM2+ Motherboard (£56.34 @ Aria PC)
Memory: Crucial 4GB (1 x 4GB) DDR3-1600 Memory (£16.99 @ Aria PC)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£39.98 @ Scan.co.uk)
Video Card: EVGA GeForce GTX 750 Ti 2GB Video Card (£89.98 @ Amazon UK)
Case: BitFenix Comrade ATX Mid Tower Case (£28.43 @ CCL Computers)
Power Supply: SeaSonic ECO 430W 80+ Bronze Certified Fully-Modular ATX Power Supply (£29.99 @ Scan.co.uk)
Monitor: AOC G2460PQU 144Hz 24.0" Monitor (£199.99 @ Amazon UK)
Total: £515.69
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-21 18:06 BST+0100

PCPartPicker part list: http://uk.pcpartpicker.com/p/GVpWQ7
Price breakdown by merchant: http://uk.pcpartpicker.com/p/GVpWQ7/by_merchant/

CPU: AMD FX-6350 3.9GHz 6-Core Processor (£89.94 @ Aria PC)
Motherboard: Gigabyte GA-970A-DS3P ATX AM3+ Motherboard (£51.00 @ Aria PC)
Memory: Crucial 4GB (1 x 4GB) DDR3-1600 Memory (£16.99 @ Aria PC)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£39.98 @ Scan.co.uk)
Video Card: Gigabyte GeForce GTX 960 2GB Video Card (£139.90 @ Aria PC)
Case: BitFenix Comrade ATX Mid Tower Case (£28.43 @ CCL Computers)
Power Supply: SeaSonic ECO 430W 80+ Bronze Certified Fully-Modular ATX Power Supply (£29.99 @ Scan.co.uk)
Monitor: AOC G2460PQU 144Hz 24.0" Monitor (£199.99 @ Amazon UK)
Total: £596.22
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-21 18:15 BST+0100

Of course! 600 quids is more than enough to make a solid gaming build.

It's squeaked slightly over budget but has an i3 processor. The motherboard is the V2 so you shouldn't have to worry about BIOS updates. The HDD is a clean pull but does have a 12 month warranty and has been removed by "IT professionals" at Aria. The Windows 7 will be eligible for the free Windows 10 upgrade so you save some money taking that route.

I notice some of the builds above don't come with Windows. If you have a copy already then you'll be able to use a better graphics card than the 750ti. If you're a student you can get Windows at a discounted price so that's also something to consider.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4170 3.7GHz Dual-Core Processor (£88.92 @ Aria PC)
Motherboard: MSI H81M-E35 V2 Micro ATX LGA1150 Motherboard (£34.00 @ CCL Computers)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory (£33.27 @ Amazon UK)
Storage: Hitachi Ultrastar 750GB 3.5" 7200RPM Internal Hard Drive (£25.99 @ Aria PC)
Video Card: EVGA GeForce GTX 750 Ti 2GB Video Card (£89.98 @ Dabs)
Case: Thermaltake Versa H22 ATX Mid Tower Case (£26.70 @ CCL Computers)
Power Supply: XFX TS 550W 80+ Gold Certified ATX Power Supply (£37.99 @ Amazon UK)
Operating System: Microsoft Windows 7 Home Premium SP1 OEM (64-bit) (£67.95 @ Ebuyer)
Monitor: AOC G2460PQU 144Hz 24.0" Monitor (£199.99 @ Amazon UK)
Total: £604.79
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-22 13:35 BST+0100

Went a little bit over but looks nice

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4170 3.7GHz Dual-Core Processor (£88.92 @ Aria PC)
Motherboard: Gigabyte GA-B85-HD3 ATX LGA1150 Motherboard (£68.30 @ Amazon UK)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory (£41.11 @ Amazon UK)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£38.70 @ Amazon UK)
Video Card: PowerColor Radeon R9 380 4GB PCS+ Video Card (£160.09 @ Ebuyer)
Case: Thermaltake Versa H22 Window ATX Mid Tower Case (£26.94 @ Aria PC)
Power Supply: EVGA 600W 80+ Certified ATX Power Supply (£34.98 @ Aria PC)
Operating System: Microsoft Windows 7 Home Premium SP1 OEM (64-bit) (£67.95 @ Ebuyer)
Monitor: Samsung S22C200B 21.5" Monitor (£97.60 @ Amazon UK)
Total: £624.59
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-22 14:29 BST+0100

It is a nice build, I think the OP wants a 120Hz monitor though and I don't think that Samsung one is. If it is however I'll be getting one myself as that would be an awesome price for 120Hz!!

That monitor is still solid for the price!