Case: Cooler Master HAF 912 - $59.99
PSU: Seasonic S12 II 620W - $84.99
Motherboard: Gigabyte GA-H77-DS3H - $99.99
CPU: 3.1GHz Intel Core i5-3450 - $199.99
RAM: 8GB G.Skill Ripjaw X 1600Mhz 1.5V - $46.99
HD: 1TB Seagate Barracuda ST - $99.99
Video Card: Sapphire Radeon HD 7850 - $249.99

Total: $860.92
This. Copied it off of some other website(tomshardware). It looks good, the cpu is almost much better than anything amd brings out. 8 gb is enough I think, you don't do any professional video or photo editing I guess? Or would you prefer a small computer?

http://www.newegg.com/Product/Product.aspx?Item=N82E16811119233 Cooler Master HAF 912 $59.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16814202006 Sapphire Radeon HD 7950 $289.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16817139028 Corsair Builder Serie CX600 $69.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16820145345 Corsair Vengeance 8 gb $54.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131837 ASUS P8Z77-V LK LGA 1155 Intel Z77 $149.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116504 Intel Core i5-3570K Ivy Bridge 3.4GHz (3.8GHz Turbo) $229.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16822148840 Seagate Barracuda ST1000DM003 1TB 7200 RPM $79.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16827135247 ASUS Black 12X BD-ROM 16X DVD-ROM 48X CD-ROM $59.99

Total: $994.92

This is the computer I would buy if I were you. ( I looked this all up on newegg.) Click on the links, check the reviews if you want. I also don't know if you want a blu-ray player?

I built this computer a few years ago and its still good enough to run most games on High settings, compose and design art using the latest adobe programs and watching anime...although my TV is only 720p but that is fine with me. I agree with above that micro atx....gawd no and never do integrated vid cards.

This build probably has a few more years left on it before a major upgrade is needed.....I could probably use more ram though.

Windows 7 Ultimate 64bit
Sunbeam Transformer IC-TR-US-BA-WOPSU Black Steel ATX Full Tower Computer Case
GIGABYTE GA-MA790XT-UD4P AM3 DDR3 AMD 790X ATX AMD Motherboard
MSI N9600GT-MD1G GeForce 9600 GT 1GB 256-bit GDDR3 PCI Express 2.0 x16 HDCP Ready SLI Support Video Card
KINGWIN ABT-730MM 730W ATX 12V Ver.2.91 SLI Ready CrossFire Ready Active PFC Power Supply
AMD Phenom II X4 955 Black Edition Deneb 3.2GHz Socket AM3 125W Quad-Core Processor HDZ955FBGIBOX
G.SKILL 4GB (2 x 2GB) 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) Dual Channel Kit Desktop Memory Model F3-12800CL9D-
Western Digital WD Blue WD5000AAKS 500GB 7200 RPM SATA 3.0Gb/s 3.5" Internal Hard Drive -Bare Drive
SAMSUNG Black 22X DVD Burner Black SATA Model SH-S223B - OEM

Don't forget the internet connection speed!

To be honest, you really don't need to pay that much for a fast computer...everything is so fast now that unless you do graphic design or 3d stuff you will never use all the capability of your computer. I have a pretty good budget rig and never had problem and I paid 1100$ for it including shipping.

Recently I just built a PC for my dad, it cost me about 600$ and it's a freaking bomb. i3 3rd gen @ 3.3Ghz is really fast, RAM is just cheap, good mobo, a graphic card if you need one, I didn't put one in my dad's computer since he only does work on his computer, but I was amaze by the mother board graphic power.
I think if you want to stream, make sure you have a good connection