Streaming Twitch will be almost completely reliant on your internet speed. If you don't have at least a 3 - 4 Mb/s upload speed you won't be able to do it.

That said, here's a build I put together for you. I assumed that you would need new everything (monitor, mouse/keyboard, Windows, etc..) so if you already have some parts you're planning to reuse let me know and we can probably bump this up a bit and make it better :)

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ TigerDirect)
Motherboard: ASRock 970 EXTREME4 ATX AM3+ Motherboard ($99.49 @ Newegg)
Memory: Crucial Ballistix Sport 8GB (1 x 8GB) DDR3-1600 Memory ($66.99 @ Microcenter)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Video Card: XFX Radeon HD 7870 2GB Video Card ($165.66 @ Newegg)
Case: Rosewill CHALLENGER ATX Mid Tower Case ($39.99 @ Newegg)
Power Supply: Corsair CX 600W 80+ Bronze Certified Semi-Modular ATX Power Supply ($49.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($82.99 @ NCIX US)
Monitor: Dell E2213H 60Hz 21.5" Monitor ($101.99 @ Staples)
Keyboard: Logitech MK120 Wired Slim Keyboard w/Optical Mouse ($12.98 @ OutletPC)
Total: $790.05
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-01-14 10:35 EST-0500)