I've been using TPU cases for a while now, and since they can be had on the cheap, I don't get too concerned on what I use. So...
On the outside, not unusual to use multi-purpose cleaner and paper towel to get off the oils, schmutz that can get on the back. Inside, like others have posted, canned air and lint-free cloth to get out anything that snuck in. Will use lens cleaner on the phone back and the inside of the case.