Edit: sorry for incoming wall of text!
that is one hefty budget and you will be able to get a good PC no matter what path you take.
that being said, most people here will say that building your own is the way to go, not surprising as Tom's Hardware is a computer hardware enthusiast site after all, and most of the time it is true, the labor costs are important to many people and is a part that is easily saved (by building your own)
HOWEVER, I do recommend that you look at certain coupon/deal sites and browse their forums for a while (somewhere like slickdeals.net or something), once in a while you'll find a hefty coupon for a nice pre-built Dell or HP or some other computer retailer. For example, a few weeks ago there was a coupon for 30% off any Dell/Alienware refurbished computer above $800, which is absolutely amazing in terms of value, a builder cannot beat that price without waiting a few months in between parts to get the best deals, I'm not sure if a builder can beat it at all actually.
Another option is custom gaming PC sites such as IBuyPower or CyberPowerPC, but for these sites, unlike Dell or HP, they will unlikely to ever offer you a coupon deal or hefty discounts, since they only sells PCs and accessories essentially, while Dell and HP and the like make money primarily off of services. So for these sites you get hefty labor costs, but the good thing is that you won't have to build it yourself, you can pay for them to overclock the PC for you, and it will be guaranteed to work out of the box. Keep in mind that for your budget the labor costs will be around $300+ compared to building your own computer with parts.
If you want to build a PC, I recommend using this site: http://pcpartpicker.com/
it is a very handy resource to keep track of which parts you want, what they cost, and how compatible they are.
As for gaming parts themselves:
For CPU Intel is often better, they have strong processors that handily beat out AMD in single threaded apps (which most games are, though that is starting to change a bit), but they are expensive and usually used in 1000+ dollar builds, which is no problem for you so I'd recommend a nice i5 4670k, hefty single core performance (on par with the i7's), can be overclocked, only loses in multi threaded apps because no hyperthreading.
GPU you can go with either AMD or Nvidia, both are superstars honestly and both pump out good cards, just look for the good brands such as Sapphire, EVGA, Asus, MSI, etc
a sample build for you would be:
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: Intel Core i5-4670K 3.4GHz Quad-Core Processor ($200.49 @ SuperBiiz)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: ASRock Fatal1ty Z87 Killer ATX LGA1150 Motherboard ($119.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($68.00 @ Newegg)
Storage: Samsung 840 EVO 120GB 2.5" Solid State Disk ($89.99 @ Amazon)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($58.98 @ OutletPC)
Video Card: MSI GeForce GTX 770 2GB Video Card (2-Way SLI) ($329.99 @ Newegg)
Video Card: MSI GeForce GTX 770 2GB Video Card (2-Way SLI) ($329.99 @ Newegg)
Case: Rosewill BlackHawk ATX Mid Tower Case ($86.99 @ Amazon)
Power Supply: SeaSonic X Series 850W 80+ Gold Certified Fully-Modular ATX Power Supply ($140.29 @ SuperBiiz)
Optical Drive: LG WH14NS40 Blu-Ray/DVD/CD Writer ($54.99 @ Microcenter)
Operating System: Microsoft Windows 8.1 - OEM (64-bit) ($98.98 @ OutletPC)
Total: $1608.66
(Prices include shipping, taxes, and discounts when available.)
powerful processor, strong mobo, GTX 770 in SLI for mind numbing performance. You can overclock if you want to, or you can leave it at stock, I noted that you aren't a huge fan of it and that's totally fine, but when the time comes that you feel like you may want to take a shot at it this is an option
the same PC will cost you around 2,300 or more on CyberPowerPC, so building it yourself is definitely something that you'd want to do!
Of course this is just a build I threw together really quick, if you tell me some more specifics on what you're looking for I could possibly cut costs/direct money better!