AMD vs Intel

budget build favor AMD.

Actually it depends more on what GPU you use. For gaming you want at minimum a Radeon 7850, or a GTX 650TI Boost. Pair that with an FX-4350 or low end i3, and you won't have a problem running games at decent settings.

G-unit has a good summary. For games, the graphics card matters much more than the CPU. The longevity of a rig built with either an AMD or an Intel CPU is more likely going to be determined by the overall quality of the parts you use, especially the PSU and to a lesser degree the motherboard.
Edit: I mostly agree with MysteryINC too, except I'm less optimistic about the APU. If you're going to get a discrete graphics card anyway, you can get a stronger i3 CPU to pair with it for about the same cost as a mid-range APU. The APU will have to be overclocked to reach the i3, which requires another $30 for a modest cooler, and likely a little more for a better motherboard (e.g. with heatsinks on its VRMs). Read the latest $350 Bonus SBM build article for an example of APU performance in games.

^1 True that. If you have a low-end GPU then you won't get more then 20 to 30 fps in any game. New PC games like BF4, NFS Most Wanted 2012 need a good GPU so they won't bottleneck the CPU and give you a pleasent experience.

Edit: If you pair the APU with something like the 7750 then the performance boost is incredible.

Yeah exactly, you could have a 3960X in your rig, but if you pair it with a poor GPU, you're going to get poor frame rates. Although I don't think anyone in their right mind would pair a 3960X with a Radeon 7790. :lol:

Unless you use it for photo editing and then the GPU doesn't really matter

Well, the OP did specify gaming, so...

Do consider balance though. Some games (e.g. online multiplayer) benefit from a stronger CPU, experiencing nasty lag spikes when there's a lot going on. Single-player games with lots of eye candy (e.g. Tomb Raider w/TressFX turned on) need a very powerful graphics card to not stutter. Check benchmarks of your preferred or intended titles, and keep in mind that lowering a setting or two can make a big difference in FPS without making the visuals look crappy.

Well,

He does want help so...

My rig: http://uk.pcpartpicker.com/p/1KQ8R
PSU: http://www.amazon.co.uk/OCZ-OCZ750FTY-UN-Fatal1ty-Series-High-Performance/dp/B003HAFZSM/ref=sr_1_3?ie=UTF8&qid=1380918680&sr=8-3&keywords=ocz+750w

The total of this build was - £816.6

If this is too high you can change the MoBo. I chose the Sabertooth as it is great for overclocking.

Not true either, rendering does require a ton of CPU horsepower, but so does the GPU. If you use CS5/6 and a bunch of custom plug ins then you could also require a pretty hefty GPU on top of that, as the rendering times will slow to a crawl if you don't have a good one.

If you have a budget of £820 for a gaming rig this is what I would do:

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-4670K 3.4GHz Quad-Core Processor (£167.99 @ Aria PC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler (£23.98 @ Scan.co.uk)
Motherboard: ASRock Z87 Extreme4 ATX LGA1150 Motherboard (£120.14 @ Amazon UK)
Memory: Mushkin Blackline 8GB (2 x 4GB) DDR3-1600 Memory (£37.46 @ Amazon UK)
Storage: Samsung 840 EVO 120GB 2.5" Solid State Disk (£73.99 @ Amazon UK)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£47.89 @ Scan.co.uk)
Video Card: EVGA GeForce GTX 760 2GB Video Card (£203.99 @ Ebuyer)
Case: Corsair 300R ATX Mid Tower Case (£59.90 @ Amazon UK)
Power Supply: SeaSonic S12II 620W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply (£71.78 @ Amazon UK)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer (£12.98 @ Novatech)
Total: £820.10
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-10-04 22:04 BST+0100)

Since you already have nice premium builds. Here's a budget build:

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-3350P 3.1GHz Quad-Core Processor (£130.00 @ Ebuyer)
Motherboard: MSI B75MA-E33 Micro ATX LGA1155 Motherboard (£41.55 @ Amazon UK)
Memory: Corsair XMS3 4GB (2 x 2GB) DDR3-1600 Memory (£24.13 @ Amazon UK)
Storage: Hitachi Ultrastar 1TB 3.5" 7200RPM Internal Hard Drive (£35.99 @ Aria PC)
Video Card: MSI Radeon HD 7950 3GB Video Card (£173.98 @ Scan.co.uk)
Case: Zalman ZM-T2 MicroATX Mini Tower Case (£19.14 @ Dabs)
Power Supply: Corsair CX 430W 80 PLUS Bronze Certified ATX12V Power Supply (£38.39 @ Amazon UK)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive (£11.10 @ Scan.co.uk)
Total: £474.28
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-10-05 12:28 BST+0100)

Not counting OS, I think $700 is easily enough. I created a test build a few weeks ago I call "sufferable," because it has the minimum level of parts I'd personally like to have, and it came in at $679. looking it over just now, I think some prices have changed a little, possibly a little lower overall.
Edit: I just made another pass through it, and it is now down to $647. There are three parts of this personal "sufferable" build with which many others will take issue. It "only" has a HD7770 in it, but includes a 128GB SSD, and the data drive is only 500GB. Those focused on games will readily forgo the SSD and put its $100 toward beefier graphics and a bigger PSU. Someone with lots of pics or video will want a bigger data drive. Some would point out that I could save $20 on the case, but I want front USB3.0 and some fans. I'd prefer some stronger parts too, but for my needs, this is what I'd consider "sufferable."

Intel Core i3-3240 Ivy Bridge 3.4GHz LGA 1155 55W Dual-Core Desktop Processor ...
Model #: BX80637i33240
Item #: N82E16819116773
$119.99 $119.99

GIGABYTE GV-R777OC-1GD Rev2.0 Radeon HD 7770 GHz Edition 1GB 128-bit GDDR5 PCI Express 3.0 x16 HDCP Ready CrossFireX Support ...
Model #: GV-R777OC-1GD Rev2.0
Item #: N82E16814125435
$99.99 $99.99

SAMSUNG 840 EVO MZ-7TE120BW 2.5" 120GB SATA III TLC Internal Solid State Drive (SSD)
Model #: MZ-7TE120BW
Item #: N82E16820147247
$99.99 $99.99

G.SKILL Value Series 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10600) Desktop Memory Model F3-10600CL9D-8GBNT
Model #: F3-10600CL9D-8GBNT
Item #: N82E16820231422
$69.99 $69.99

LITE-ON DVD Burner - Bulk Black SATA Model iHAS124-04 - OEM
Model #: iHAS124-04
Item #: N82E16827106289
$17.99 $17.99

Rosewill CHALLENGER-U3 Black Gaming ATX Mid Tower Computer Case ,comes with Three Fans-1x Front Blue LED 120mm Fan, 1x Top ...
Model #: CHALLENGER-U3
Item #: N82E16811147060
$69.99 -$10.00 Instant $59.99

ASRock H77 Pro4/MVP LGA 1155 Intel H77 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard
Model #: H77 Pro4/MVP
Item #: N82E16813157302
$79.99 -$5.00 Instant $74.99

Antec EarthWatts Green EA-380D Green 380W Continuous power ATX12V v2.3 / EPS12V 80 PLUS BRONZE Certified Active PFC Power ...
Model #: EA-380D Green
Item #: N82E16817371033
$59.99 -$15.00 Instant $44.99

Western Digital WD Blue WD5000AAKX 500GB 7200 RPM 16MB Cache SATA 6.0Gb/s 3.5" Internal Hard Drive Bare Drive - OEM
Model #: WD5000AAKX
Item #: N82E16822136769
$59.99 $59.99

Subtotal: $647.91 (Not counting a few promo codes currently in effect, but also not counting shipping and extortion)