Is my processor going to hold my graphics card back?

I just got a Geforce GTX 770 and my processor is an AMD Athlon II X4 640 processor.

You need to remember that computing today is highly graphics oriented so the more capable your graphics card, the more graphics tasks the CPU can hand off to the graphics card and the faster the card can complete those tasks. The CPU pretty much just hands off the tasks to the GPU, and it takes very little CPU horsepower to hand off tasks. I would be more concerned with having an ample amount of RAM and free disk space before worrying about your CPU - and, of course, enough PSU power to support that new card.