Why would you build an AMD rig on a $1k-2k budget? I don't understand. I'd honestly recommend Intel at this budget, but I'll give you some options to work with.
INTEL Build:
[url=http://pcpartpicker.com/p/Wg67]PCPartPicker part list[/url] / [url=http://pcpartpicker.com/p/Wg67/by_merchant/]Price breakdown by merchant[/url] / [url=http://pcpartpicker.com/p/Wg67/benchmarks/]Benchmarks[/url]
[b]CPU:[/b] [url=http://pcpartpicker.com/part/intel-cpu-bx80637i73770k]Intel Core i7-3770K 3.5GHz Quad-Core Processor[/url] ($320.94 @ Amazon)
[b]CPU Cooler:[/b] [url=http://pcpartpicker.com/part/nzxt-cpu-cooler-havik140]NZXT HAVIK 140 90.3 CFM CPU Cooler[/url] ($59.98 @ Outlet PC)
[b]Motherboard:[/b] [url=http://pcpartpicker.com/part/asrock-motherboard-z77extreme4]ASRock Z77 Extreme4 ATX LGA1155 Motherboard[/url] ($134.99 @ Amazon)
[b]Memory:[/b] [url=http://pcpartpicker.com/part/gskill-memory-f31866c9d16gxm]G.Skill Ripjaws X Series 16GB (2 x 8GB) DDR3-1866 Memory[/url] ($110.50 @ Newegg)
[b]Storage:[/b] [url=http://pcpartpicker.com/part/a-data-internal-hard-drive-asx900s3256gmc]A-Data XPG SX900 256GB 2.5" Solid State Disk[/url] ($199.99 @ Newegg)
[b]Video Card:[/b] [url=http://pcpartpicker.com/part/gigabyte-video-card-gvr797oc3gd]Gigabyte Radeon HD 7970 3GB Video Card[/url] ($399.99 @ Newegg)
[b]Case:[/b] [url=http://pcpartpicker.com/part/corsair-case-c70arcticwhite]Corsair C70 Arctic White (White/Black) ATX Mid Tower Case[/url] ($109.99 @ Newegg)
[b]Power Supply:[/b] [url=http://pcpartpicker.com/part/xfx-power-supply-p1850bnlg9]XFX ProSeries 850W 80 PLUS Silver Certified ATX12V / EPS12V Power Supply[/url] ($119.99 @ Newegg)
[b]Optical Drive:[/b] [url=http://pcpartpicker.com/part/lite-on-optical-drive-ihas124-04]Lite-On iHAS124-04 DVD/CD Writer[/url] ($17.98 @ Outlet PC)
[b]Total:[/b] $1474.35
[i](Prices include shipping, taxes, and discounts when available.)[/i]
[i](Generated by PCPartPicker 2013-05-10 00:40 EDT-0400)[/i]
AMD Build:
[url=http://pcpartpicker.com/p/Wg80]PCPartPicker part list[/url] / [url=http://pcpartpicker.com/p/Wg80/by_merchant/]Price breakdown by merchant[/url] / [url=http://pcpartpicker.com/p/Wg80/benchmarks/]Benchmarks[/url]
[b]CPU:[/b] [url=http://pcpartpicker.com/part/amd-cpu-fd8350frhkbox]AMD FX-8350 4.0GHz 8-Core Processor[/url] ($189.55 @ Amazon)
[b]CPU Cooler:[/b] [url=http://pcpartpicker.com/part/nzxt-cpu-cooler-havik140]NZXT HAVIK 140 90.3 CFM CPU Cooler[/url] ($59.98 @ Outlet PC)
[b]Motherboard:[/b] [url=http://pcpartpicker.com/part/asrock-motherboard-990fxextreme4]ASRock 990FX Extreme4 ATX AM3+ Motherboard[/url] ($139.99 @ Amazon)
[b]Memory:[/b] [url=http://pcpartpicker.com/part/gskill-memory-f31866c9d16gxm]G.Skill Ripjaws X Series 16GB (2 x 8GB) DDR3-1866 Memory[/url] ($110.50 @ Newegg)
[b]Storage:[/b] [url=http://pcpartpicker.com/part/a-data-internal-hard-drive-asx900s3256gmc]A-Data XPG SX900 256GB 2.5" Solid State Disk[/url] ($199.99 @ Newegg)
[b]Video Card:[/b] [url=http://pcpartpicker.com/part/gigabyte-video-card-gvr797oc3gd]Gigabyte Radeon HD 7970 3GB Video Card[/url] ($399.99 @ Newegg)
[b]Case:[/b] [url=http://pcpartpicker.com/part/corsair-case-c70arcticwhite]Corsair C70 Arctic White (White/Black) ATX Mid Tower Case[/url] ($109.99 @ Newegg)
[b]Power Supply:[/b] [url=http://pcpartpicker.com/part/xfx-power-supply-p1850bnlg9]XFX ProSeries 850W 80 PLUS Silver Certified ATX12V / EPS12V Power Supply[/url] ($119.99 @ Newegg)
[b]Optical Drive:[/b] [url=http://pcpartpicker.com/part/lite-on-optical-drive-ihas124-04]Lite-On iHAS124-04 DVD/CD Writer[/url] ($17.98 @ Outlet PC)
[b]Total:[/b] $1347.96
[i](Prices include shipping, taxes, and discounts when available.)[/i]
[i](Generated by PCPartPicker 2013-05-10 00:42 EDT-0400)[/i]
There's no need to put a "high-end" all in one water cooler in your build. The NZXT Havik 140 will do a great job at cooling. Check out Logan's review. He even said it performs equal to and occasionally outperforms his H80.
The HD 7970 is one of the best GPUs $400 can buy. I'd recommend it over the GTX 680 due to price/performance. GTX 680s usually run around $500+.
I also gave you the option of getting the i7 3770k and the FX 8350. The i7 3770k typically outperforms the FX 8350, but it also costs quite a bit more. Just choose whichever one you think will get the job done. From what I've read in the comments of this thread, it sounds as if the FX 8350 would suit your needs better.
If you have any questions, feel free to ask. I'll do my best to get back to you, but don't be surprised if it's a couple of days before I do. Keeping track of who has replied to what is really difficult on this site.