Is there a computer I can buy that can run these game or make my own? If I was going to make my own I would probably want to spend around either 600-700 US Dollars.
At that price point it's going to be tough to get a gaming computer. Fiddling around on Cyberpower I got a pretty decent rig for about $800 with monitor, but that's outside your budget. Your best bet may be to go find a $500 special from Best Buy, get a cheapish (~$120) monitor and buy a $80 video card to add in. You'd be surprised how much gaming you can get done with a $80 video card.
Some things to consider:
don't fall for the lure of six or eight core processors. You don't need them.
For gaming, you won't use more than four physical cores since parallelism is hard.
If you're using Intel CPU, use NVidia graphics cards. If you use AMD CPU, use AMD/ATI graphics cards. Yes, you can make them work the other way around, but why swim upstream?
You don't need an LCD monitor bigger than 19" but you really do want to get one that supports 1920x1080 resolution, otherwise why bother with a nice computer in the first place?
You don't need the fastest RAM out there, but you probably do need at least 6 GB of it. That shouldn't be a problem on most desktops.