Hey man, i've made a list of parts for you;
PCPartPicker part list:
http://pcpartpicker.com/p/2lQt0
Price breakdown by merchant: http://pcpartpicker.com/p/2lQt0/by_merchant/
Benchmarks: http://pcpartpicker.com/p/2lQt0/benchmarks/
CPU: Intel Core i7-4770K 3.5GHz Quad-Core Processor ($319.99 @ Newegg)
CPU Cooler: Corsair H100i 77.0 CFM Liquid CPU Cooler ($89.99 @ NCIX US)
Motherboard: MSI H87M-G43 Micro ATX LGA1150 Motherboard ($88.78 @ NCIX US)
Memory: Mushkin Stealth 16GB (4 x 4GB) DDR3-1600 Memory ($154.99 @ Newegg)
Storage: Samsung 840 EVO 120GB 2.5" Solid State Disk ($89.99 @ B&H)
Storage: Seagate Barracuda 3TB 3.5" 7200RPM Internal Hard Drive ($104.99 @ NCIX US)
Video Card: Gigabyte GeForce GTX 770 4GB Video Card ($383.99 @ SuperBiiz)
Case: Corsair 350D MicroATX Mid Tower Case ($79.99 @ Microcenter)
Power Supply: Silverstone Strider Gold Evolution 750W 80+ Gold Certified Fully-Modular ATX Power Supply ($152.98 @ Newegg)
Total: $1465.69
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-17 14:36 EST-0500)
That should do what you want very well, with ample room for future expansion. The i7 will help with encoding of the frames for streaming, the 770 that'll give you great performance in games. SSD to store a few games and OS, 3TB HDD for mass storage all in all, it's something i'd buy if i had your budget.