Will this achieve 60fps on most games?

Everything looks fine. I would up the GPU a bit though if you're willing to spend the money. GTX 660ti, 670 or something similar would do really well.

do you know if it will all fit in the case?

With that kind of budget and just a 660 gtx, isn't good, get a 670, drop the aio watercooler, i'd get a ssd with mlc nand over tlc nand.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-3570K 3.4GHz Quad-Core Processor (£161.99 @ Aria PC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler (£24.99 @ Amazon UK)
Motherboard: ASRock Z77 Extreme4 ATX LGA1155 Motherboard (£101.98 @ Dabs)
Memory: Corsair XMS3 8GB (2 x 4GB) DDR3-1600 Memory (£32.99 @ Amazon UK)
Storage: Western Digital RE3 1TB 3.5" 7200RPM Internal Hard Drive (£44.34 @ Aria PC)
Storage: Sandisk Ultra Plus 256GB 2.5" Solid State Disk (£130.97 @ Amazon UK)
Video Card: Gigabyte GeForce GTX 670 2GB Video Card (£299.99 @ Aria PC)
Case: Corsair 300R ATX Mid Tower Case (£59.99 @ Amazon UK)
Power Supply: XFX 750W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply (£72.17 @ Aria PC)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£12.98 @ Scan.co.uk)
Keyboard: Microsoft Sidewinder X4 (UK Layout) Wired Standard Keyboard (£32.99 @ Novatech)
Other: Logitech G9X Call of Duty Laser Mouse @ amazon.co.uk (£40.90)
Total: £1016.28
(Prices include shipping, taxes, and discounts when available.)

The xfx psu is made by seasonic, it's a quality psu, it will do even for two 670 gtx if you plan to add another one, the psu has the amps for it and the motherboard supports it.

hey thanks django! Can you clarify some stuff for me mate:

What is an AIO watercooler? And what is MLC and TLC Nand?

http://www.ebuyer.com/368650-evga-nvidia-gtx670-915mhz-...

this instead of the 660 then

All in one water cooler, i'd get an aircooler over the Antec Kuhler, depending on the air cooler you'll get even better temperatures for less money and without any leaks risk.
MLC means multi level cell, TLC means triple level cell.
You'll get slower speeds with tlc nand, specially under heavy load, overall tlc offers less performance and endurance.
That 670 gtx is good, but i rather get the 670 gigabyte windforce 3x, it has a better cooling system.

won't that windforce be loud as a jet though? My plan is to make a quiet system using the obsidian 550d

Nope, it's a quiet card, the blower style coolers are more louder, also if you want a silent case get the fractal design define R4, it's a much better case than the obsidian 550D.
Here's the updated list with the fractal case : http://uk.pcpartpicker.com/p/END4

When I built my 2013 intel/Nvidia rig I got a Gigabyte GTX 670 and I can tell you I have not had any problems with the fans most of the time I can't even hear them running. As for the CPU cooler I went for the Corsair H100i and am really happy with it. I started with the stock heatsink/fan and not only did it sound like a jet engine but it did not seem to cool very good ether.

I choose the same CPU as you did and it is working out really good. I have been with AMD since the K6-2 so there was a little learning curve for me going back to intel. I have also been with ATI since my Geforce 7800GS(AGP) died.

Anyway this is the rig I just built
Motherboard: Asus Z77 Sabertooth
CPU : i5 3570K
Ram : G Skill 16gig DDR3 1600
SSD : OCZ Vertex 4 2.5" 128GB SATA III
CPU Cooler : Corsair H100i
Case : In Win Grone White 5 120mm white led fans
Even with 7 120mm fans plus the 2 fans on my GPU I can't really hear the system running unless the house is quiet.