1st Gaming Pc budget of £750/$1100

That is a challenge, if you're needing everything for £750. As you're needing peripherals as well, it does reduce what we can do with the PC itself. However, have a look here. The keyboard and mouse will get you started, and the monitor has good reports. The CPU is a dual-core one...I'd ideally go for quad core but the budget isn't there (just make sure you get the retail one, not OEM, as the latter doesn't have a cooler with it).

What programs? All you really need is Open Office and AVG Free - both of which are excellent free programs that you can use. There's no need to pay money for a full license of MS Office. I wouldn't recommend getting expensive peripherals now. Just get cheap stuff like a Logitech MK320.

Try this:

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-3350P 3.1GHz Quad-Core Processor (£139.49 @ Dabs)
Motherboard: ASRock Z75 Pro3 ATX LGA1155 Motherboard (£66.95 @ Amazon UK)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory (£45.99 @ Amazon UK)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£55.00 @ Amazon UK)
Video Card: XFX Radeon HD 7870 2GB Video Card (£166.07 @ Aria PC)
Case: Corsair 200R ATX Mid Tower Case (£46.70 @ Amazon UK)
Power Supply: SeaSonic S12II 620W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply (£76.88 @ Amazon UK)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£11.79 @ Amazon UK)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) (£68.04 @ CCL Computers)
Total: £676.91
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-03-20 22:34 GMT+0000)

Are you looking for part advice, or advice on how to build your own PC? I can provide some links that show how easy a PC build has gotten.

Regardless, here is a list of parts you will need to buy:

CPU: I would always pick this component first. Intel i5 3350P is the sweet spot if you don't plan on overclocking, and the Intel i5 3570K is the sweet spot if you do plan on overclocking at some point in the future.

Motherboard: Pick one appropriate for your CPU and has a PCI-e slot for your GPU.

Graphics Processor Unit (GPU): This is component usually has the biggest influence on games graphics performance, though there are games that are "CPU bound." Regardless, buying an expensive GPU but a cheap CPU will leave performance CPU bound, so the key is to find a balanced system.

Case: Read some reviews. Cases are easily the most "future proof" of any PC component, so feel free to splurge a little and buy a case you think both looks nice and has good reviews for being easy to work with. I can recommend the Corsair 500R since its what I own, but you have a lot of options.

RAM: I can already tell you that the sweet spot is 8 gigabytes of 1600 MHZ DDR3 RAM from a reputable maker like Corsair. 16 gigabytes is overkill unless you have very specific programming needs, and 4 gigabytes is only a little cheaper than 8.

Storage: This will get a little complicated because you have two options: A Solid State Drive (SSD) which drastically will reduce load and boot times, but comes with substantially less space for the money (~60-256 gigabytes). Or, you can by a traditional Hard Drive (HD) which is slower, but is slightly cheaper and has a lot more space (500-2,000 gigabytes). Or, you could get one of each, which I think is a very realistic option with your budget.

Disc Reader: Some people actually build PCs without these, but I don't think that time has come yet. Plan to spend $20-$40 for a CD/DVD writer/reader.

Power Supply Unit (PSU): Use a calculator like this (http://extreme.outervision.com/psucalculatorlite.jsp) to figure out how many Watts you will need based on your chosen components above. Buy a PSU from a reputable company (Antec, Corsair, SeaSonic, Silverstone, others) with your needed Watts.

Operating System (OS): Windows 7 or Windows 8. I prefer 7, but they both cost the same, ~$100, and are almost identical for gaming. You only need the basic Home edition.

Monitor: I recommend you go to an electronic store and see what sort of monitor size you would be comfortable with and go from there. Just realize that higher resolution monitors require substantially more powerful hardware to run. A basic 1920x1080 monitor (which I think is what most gamers have) will run most games with the $200 card on high or ultra settings and keep a high frame rate. A "4k" monitor displays four times the number of pixels and as such needs a lot more juice, which is why people build PCs with 2 $300+ GPUs.

Mouse: Buy a Logitech wireless mouse. They are great for gaming and just great mice in general.

Keyboard: Doesn't really matter much. I would say don't buy any expensive keyboard unless it features mechanical keys. Otherwise, just go to an electronics store and pick one that you like the look, feel, and price of. A good keyboard is unless important for gaming than a good mouse.

Sound System: I, and a lot of gamers, prefer to use headphones. I recommend you shop for a pair online. If you plan on doing any online gaming, you will want a microphone. Most buy a headset with a microphone attached.

Optional----

CPU Cooler: Every CPU comes with a fan, usually called the "stock cooler." It will work fine, provided you don't overclock the CPU (make it run faster), which means it will generate more heat and need a better cooler. For a first time builder, I would just plan on using the stock cooler and if you later want to try your hand at overclocking that will be the time to get a better one.

Sound Card: Most motherboards come with pretty good on-board sound, more than enough for a normal headset. However, if you plan on using a more expensive sound set up, it might bet worth buying a premium sound card. Not recommended for your budget unless you are a real audiophile.

While this guy is building a small, budget PC, I none the less think he does an excellent job showing how easy a PC build can be and what steps you need to take: http://www.youtube.com/watch?v=Bh455l3348s

Here you go m8
http://uk.pcpartpicker.com/p/L4zW

Agree with G-Unit. Don't spend money on an expensive keyboard and mouse for the moment.

You can save up for a better keyboard and mouse in the future but if you scrimp on parts like the CPU + GPU now you will find yourself stuck with them as they are very expensive to replace. You don't want to be looking at £200+ to upgrade your GPU in a years time...

i5 + 7870 should be enough to handle those games at high settings on a 1080p monitor. Maybe not so smooth at ultra settings, but it should handle high just fine.

Yeah especially if your budget is that tight. You don't want to blow a lot of it on an expensive keyboard and mouse. I don't even like doing that on £1500 builds.