I think the cheapest and easiest way to fix this is to simply buy some plywood, and make a simple box you can put your computer in. You'll want some kind of sound absorbing material, but don't get any expensive stuff, I would recommend those egg carton looking things some people put on their bed for comfort. (sorry don't know what to call them, but if you see them you'll know what I'm talking about).
Use that material to help absorb any sounds your PC makes, and to keep your PC from overheating it might be best to leave one plate of the box unattached, and especially don't block any air vents!
Basically if you can make something to shield away your PC sounds, that should do the trick. If you have a closet think about putting your PC in the closet while doing recording, that might be the easiest and cheapest method....although it'd require you to move your PC a lot unless u can leave it there permanently....which might be annoying if you actually use ur closet like most people lol.