Should I charge peers and superiors for doing computer work on their personal devices?

Not at all. In fact, it's an ethical violation for the officer to ask you to spend your free time performing a service that would normally be charged for. Extra wrong if it's a commander. Think about it like this, if your OIC asked the Soldiers in your section to wash his/her car, would that be OK? Absolutely not. Or if they asked the unit armorer to fix their personal gun? No
This is something that you should share with your PSG. A Senior NCO, or this officer's commander, needs to pull them aside and explain the ethics and the optics of asking enlisted to perform personal tasks. NCOs have been reduced for this, officers have received GOMORs for it, and Generals and Commanders have been relieved for it, even though they weren't intending to take advantage of anyone.

This sounds like legal, ethical and possibly moral issues that could turn very ugly.

Tell them they make enough money to take it out in town. Otherwise, my opinion is he is taking advantage of your skills. I would check with the JAG's to see if there is anything wrong if you want to do the work, and get paid for it. It seems like there is a UCMJ law against it, but I can't remember.