Yep, knowing me I'd end up frying one of the components, so I think I'd best leave it to the professionals.
I went ahead and asked a local place who build custom gaming rigs. They gave me two quotes, an OK one and a higher end one. The specs were:
OK:
I3 CPU
500GB HDD
4GB RAM
GT620 graphics card
I5 CPU
1TB HDD
8GB RAM
GT630 graphics card
All things considered, I'm thinking of going with the higher end model, perhaps supplementing it with an SSD depending on price, then maybe looking at a better CPU? saltpastillen mentioned that an I5 is still viable, and somewhat better than an I7, but I'm wondering if it's worth going a little extra? I'm probably going to spend a little extra and get a 23" monitor, which would bring the total (sans modifications) to about $1400 (windows included).