Look at car-pc websites. There are cases designed to be bolted down, and also power supplies that accept 12v input, so you don't need to be on shore power or run an inverter.
Other than that, RVs are mostly plywood on the inside - find the spot you want your PC to live, and put some lag screws through the case and into the floor (for example). Might want to look under the RV to make sure there's no water or electric lines in the way.
Build with an aluminum case so it has less inertia to bounce around while you're driving. A USB dvd+rw drive might be a good idea too, so it can be stored separately and plugged in when needed.
2013 KZ Sportsmen Classic 200, 20 ft TT
2020 RAM 1500, 5.7 4x4, 8 speed