Maximum PC

Hey all, I'd like some input on some ideas on hardware, and hope that I won't start any wars at all... just trying to get best bang and best performance on the budget.

Desire/Goal: Build a gaming rig that will play minecraft/tekkit/technic with 30+ FPS, while still being able to record screens (30+ FPS in video), and possibly live stream (480p - 1080p not a big deal which end) all at the same time.


Budget: Up to $1200 (can be changed but would like to keep it as low as possible.)

Plan on recycling: Windows OS, Mouse, Monitor (AOC 24in, 1080p, DVI/VGA connections), sound system

Need:
Case,
Mobo,
CPU,
GPU,
RAM (8GB+ is recommended),
PSU,
Cooling (preferably liquid [not needed]),
1 SSD (64 GB < ssd > 256 GB)
HDD (>= 1TB)
and any required cabling (will not count towards budget)

Not needed: Optical drive [I have a laptop that will burn/play discs if needed... but there are USB drives and external drives for that... ]

My beginning thoughts were mainly based around the GPU + CPU combo that will provide me with the power to do what I want (See Desire/goals).
I was thinking something like a Radeon HD 7850... would that be enough to do what I want? Would an nVidia card of similar price have better performance? (I am not a "fanboy" of either, so it doesn't matter to me which company I was to go with.)

Also for CPU: was thinking either a Core i5 - 25xxK (for some OC'ing) or an AMD FX-8150 chip. I know there are price differences, and power differences, and I know that Intel will beat the AMD chip too... but will what I want to do use the CPU that much over the GPU? For instance... would I need an i5 with one of the GPU's to help with my goal?


The rest can pretty much be built off of what fits within the budget I would assume. Case quality/aesthetics are not a huge matter to me. I will of course need a PSU that would support all this, and have room to breath in case I wanted to add another card in the future.


Any help will be appreciated on this matter. If you don't think it can be done within that budget, then just say so, but provide some reasons that it would need to go up and such.

Also, I do not mind where I would get the parts.. even if they are from 100+ (kidding, more like 10) retailers. But at least somewhat known retailers.