The A10-5800K is the exact same price as the Core i3-3220. Here's how they compare to one another: http://www.anandtech.com/bench/Product/675?vs=677
Given you kept that Athlon system for so long, you obviously want to build for the long haul. So with that in mind... and considering you plan to game while also recording the screen, I'm going to suggest a midrange GPU instead of the A10-5800K.
Since that is an eMachines, you're also going to need a PSU upgrade. You can reuse the RAM though luckily... I got $405 before MIRs, comes with a copy of far cry 3 as well. Link
Given the budget, I'd suggest saving for the SSD at a later date. Prices on 64GB SSDs aren't as good and ya certainly don't want to buy anything smaller in my personal opinion. But I can't agree more with Allikat, I suggest saving up $100 for a 120 or 128GB SSD... you won't believe the difference in responsiveness it will bring. Throw an SSD into it and the system will probably last you until some piece of hardware dies.