850$,First time build Gaming and streaming

I think that this build will do you well. I left some room in the budget open for shipping. Some of the merchants listed will be from NewEgg, but you can change the merchant with an option on the top right of the page.
If you have any questions, just ask.

PCPartPicker part list: http://pcpartpicker.com/p/mRaY
Price breakdown by merchant: http://pcpartpicker.com/p/mRaY/by_merchant/
Benchmarks: http://pcpartpicker.com/p/mRaY/benchmarks/

CPU: Intel Core i5-3470 3.2GHz Quad-Core Processor ($149.99 @ Microcenter)
Motherboard: ASRock Z77 Extreme4 ATX LGA1155 Motherboard ($114.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($38.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($69.99 @ NCIX US)
Video Card: Asus Radeon HD 7870 GHz Edition 2GB Video Card ($222.21 @ NCIX US)
Case: Rosewill Challenger-U3 ATX Mid Tower Case ($59.99 @ Newegg)
Power Supply: XFX 650W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply ($69.99 @ NCIX US)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($17.99 @ Newegg)
Total: $739.14
(Prices include shipping, taxes, and discounts when available.)

That's a good build but no need to get a Z77 motherboard if you're not using the 3570K. You could drop that to an H77 and then upgrade the GPU to a Radeon 7950.

Also that processor is pick-up only. And I feel that coming from Norway to an American store just to pick up a processor seems like a hassle a bit.

Yeah you'd probably pay double the system cost just for a ticket from Oslo to Santa Ana, CA.  

If you don't know a local shop where you live I'd suggest checking the link in my signature.

Here was the best I could do:

Case: Fractal Design Arc MIDI - 329 kr
PSU: Corsair CX600 V2 - 539 kr
Motherboard: Gigabyte GA-H77X-DS3H - 699 kr
CPU: 3.2GHz Intel Core i5-3470 - 1.399 kr
RAM: 8GB Corsair Vengeance RAM 1600MHz 1.5V - 329 kr
HD: 1TB Seagate Barracuda ST - 569 kr
Optical: Sony DVD Burner - 139 kr
Video Card: Sapphire Radeon HD 7850 - 1.404 kr

Total: 5.609 kr (USD $977.90)

@g-unit. Is there a good H77 board that has two pci e 3.0 slots that won't limit a graphics card. If there is, I am unaware.
@Deemo. The list was put together as what to buy, not necessarily where to buy from. That's why I included information for the OP to change the merchant.

I know I know I was just being funny  

Probably something like this: http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Your link just took me straight to the Newegg homepage. Probably something on my end, but I will take your word for it.

thanks