I'm thinking of buying an AMD Radeon HD 6790 video card and I've checked the compatibility of everything else in my desktop so far. However I'm not sure if my CPU can handle it because I have no idea on how to check or if it's even a big issue at all (I'm assuming it is). The CPU is an AMD Athlon II x4 645. Help would be super appreciated.
Actually the CPU is one of the least worries. The biggest issue is the power supply. You need a minimum 400 watt supply to effectively run this card. max draw for the HD6790 is 150 watts in 3D. That's not continuous but still. You want some headroom in the power supply.
Other then that as long as you have the slot in the motherboard for the card you should be fine. Your CPU has nothing really to do with your graphics, you have a decent CPU which is all that matters. If you had a really slow older CPU say a Sempron or something like that. Then you have to ask yourself why put that much into a graphics card for such a crappy CPU. But other then those extremes you don't have to worry. Keep in mind that a balanced system is better then over kill on just one piece of hardware.