There are guides online on how to adapt a computer PSU for desktop power supply usage but I'm not sure what that really entails. Personally I prefer using batteries because I'm more confident they won't burn my house down. A little sealed lead acid battery will do the job just fine for a plenty long time and you can charge it with a 1 amp trickle charger or something like that.
Hydrogen balloons float much better than helium balloons, since they are half the density. In fact hydrogen was the predominant gas for balloons and airships before the hindenburg....There are some other very fun things you can do with hydrogen balloons because they are flammable, but I'll leave that to your imagination. I'd recommend against using a glass jar because in case something does explode you'd probably prefer not to have glass shrapnel. I used clear acrylic tubing with some PVC elbows to make my u-tube, and added some ball valves to enable me to fill some balloons.
Hydrogen on it's own is quite volatile... hydrogen and oxygen together are extremely volatile. I'd recommend not making very big quantities. Have fun and be careful
P.S. If you want something else fun to do with your finished electrolysis rig, find a cheap fuel cell!
In fact a prefabbed fuel cell may be just what you're looking for, many of them work both ways (can perform electrolysis very efficiently if you apply charge, or generate electricity if you supply hydrogen and oxygen) I built a fuel-cell-car in high school using one of these types of kits and it was quite fun:
http://www.amazon.com/gp/product/B0..._m=ATVPDKIKX0DER&pf_rd_r=01HTFEQ5NTQT086PJ90A