CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ Amazon) - Great for editing and running Minecraft since it's a CPU intensive game.
Motherboard: MSI 970A-G43 ATX AM3+ Motherboard ($59.99 @ Newegg) - Perfect for the CPU and memory.
Memory: Kingston Predator Series 8GB (2 x 4GB) DDR3-2133 Memory ($58.61 @ NCIX US) - Enough memory to game and edit/render
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($61.98 @ Outlet PC) - Great hard drive space for Minecraft videos, photos, videos, etc.
Video Card: XFX Radeon HD 7870 2GB Video Card ($184.99 @ Newegg) - This graphics card can run Minecraft shaders no problem whilst recording
Wireless Network Adapter: Rosewill RNX-N250PCe 802.11b/g/n PCI-Express x1 Wi-Fi Adapter ($20.33 @ Amazon) - You won't need if this if you're planning to use ethernet (Highly recommend using ethernet!)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($29.99 @ Newegg) - Cool looking case
Power Supply: XFX 550W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply ($44.99 @ NCIX US) - Clean power supply - more than enough!
Optical Drive: LG GH24NS95 DVD/CD Writer ($12.98 @ Outlet PC) - For CD games and to install OS unless you are installing from a Flash drive.
You could squeeze in an SSD for Minecraft, running Minecraft on the same Hard Drive whilst recording will lag out the game, I would highly recommend you buy a second HDD or SSD!
Total: $583.85