Newbie CPU upgrade questions...

Hello there. I have a few things I wanna run by you guys. I plan on upgrading my processor but there are several questions I need answered before I can feel safe with a purchase.
First off What would be a cheap and efficient upgrade from my current AMD A6-3620 2.20 ghz CPU.
Second off I understand my motherboard may not be able to use certain high end processors(The link at the end lists available upgrades).
Just trying to get some games to play faster. Due to the belief of my processor bottle-necking my gpu or vice versa.

Game I want(wish to run on high) is far cry 3.

I know my evga geforce 560 would be up to it graphic wise...but my cpu is questionable.

Also if I wish to self install it instead of going to a tech. You think I will be safe, or should I get a pro?



Stock processor and info listedupgraded GPU and PSU)
http://h10025.www1.hp.com/ewfrf/wc/document?cc=us&lc=en&dlc=en&docname=c03135882#N412

http://www.cpubenchmark.net/cpu.php?cpu=Intel Core i5-3570K @ 3.40GHz&id=828

The graph of the bottom should help u most of those at the top are within the budget of 170-250$ but then u have to count the motherboard upgrade

The Amd fx 8350 triples the performance of ur cpu but many games dont use 8 cores...

and the Intel i5-3570k is 2 and a half better then ur cpu but is a heavy multitasker

I just don't know if a newer motherboard will fit my case(btw its a store bought I've just been upgrading). Perhaps I can just upgrade to the highest my fm1 socket can handle? It goes up to at least a quad core with 3.0 ghz I think.