I've seen Linux running on an external hard drive, but I doubt it will work on Windows. Not that you'd want to, because HDD performance will most likely suffer, especially if your external only uses USB 2.0. Really, if you can't afford a $90 hard drive you shouldn't spend almost $700 on a video card.
Btw, I looked a little at your build, and saw that Micro Center has a combo deal with your mobo and CPU. It's $85 less than buying them from the current stores, and you'll save $80 more if you get the 4670k instead, which is only mariginally slower than the 4770k.