i5 3570 k or 8350?

So I'm guessing 8350 then lol

Since you're only using a single GPU and looking to play Games, Multitask and Record Video Game Sessions (I assume) then I would recommend the AMD FX-8350. If you had two GPUs, my recommendation would be the Intel CPU.

Intel CPUs are more powerful CPUs in general. Each Intel Core does more work per clk cycle than an equivalent AMD core (Integers per Clock or IPC). Since AMD cores are paired in Modules with shared FPU resources there is a known problem with scheduling the shared resources between both cores within a module. Some people disable one core per module in order to alleviate this issue. These people tend to get a boost in gaming, believe it or not. There is also the issue of memory bandwidth which does show up in some games and is notably apparent in Dirt3, F1 2012 and other racing sims/games as seen here:
[img]http://media.bestofmicro.com/W/W/369248/original/F12012UltraCPUBottleneck2013.png[/img]

It's not as simple as "Multi tasking = 8 cores". Just saying.