How are naval jets moved around below the deck of a carrier?
As Simon pointed out, they use tugs. Obviously the engines are not started below the deck as it would have potentially deadly consequences. 

The tugs are either electric or powered by a diesel/JP-5 engine. For example, the USN uses the A/S32A-32 towing tractor (also called spotting dolly SD-2), among others.



Image from zone-five.net

This tug is powered by a diesel engine and controlled using a joystick. In order to be useful in the confines of the deck, the tug can rotate with a zero turning radius. The operation is pretty much similar to the other tugs- the tug is moved so that it lifts up the nosewheel, facilitating the aircraft movement.

The French, on the other hand use remotely controlled mototk tugs on board de Gaulle.



Image from mototok.com