First you need to realize that the term stabilizers are not for leveling, jacks are for leveling. Stabilizers are usually weaker than jacks and don't hold much weight, they just prevent movement. I think you want to ask if the Class C has leveling jacks. If you don't ask about them don't expect them to be there.
If you don't have an automatic system, the principal is the same as for trailers. First level the rig side to side with blocks or jacks, then level it front to back. If the unit does not have stabilizers (since they might not even be strong enough to hold up a Class C, then you'll need to chock the rig somehow. The big difference is that you can also use the E-brake to stabilize the rig.
Automatic levelers are not standard in a Class C, but it you get them, they would be good for both leveling and stabilizing typically with the touch of a button or two. But you might still want to consider chocking the wheels especially if the ground is not very even. If the jacks put the rig up so high the tires are not touching the ground, you probably don't need to chock them.
Read the manual carefully. If you have slides there's a good chance the rig needs to be level and stabilized to extend any slideouts. They work best when operated on the level and when the rig is stabilized to minimize unnecessary movement.
Most motorhomes have levelers, the ones that are smaller with a truck style cab (which I have 1 of in storage right now) uses butterfly jacks and we use the orange blocks. We have a few 4x4 timbers cut down to 1ft in length to use under the butterfly jacks that are mounted to each corner.
My friends class c didn't come with any kind of leveling devices. He used 2x8 under the rear wheels to get level side to side and home made ramps for the front. He bought two scissor jacks to stabilize the rig, then switched over to two bottle jacks. He said it was easier with the bottle jacks.
With some wood the bottle jacks could be used to level out the camper. Use chocks also.
Hydraulic jacks might be easier but I wouldn't trust them to hold for long periods of time, . I know a trailer would be a bad place for a hydraulic jack because it's too easy to overtorque them and possibly cause a frame problem, but I'm not at all sure how this would impact a Class C. The pad on a bottle jack is just too small.
I'm not sure what "butterfly jacks" are but I would trust permanent stabilizers (scissor, telescoping or stacker styles) that attach to the frame, then swing down to the ground more than a bottle jack.
Save the hydraulics for an automatic system that is fastened to a Class C frame than the small circular post of a hydraulic jack. Just my thoughts....