How big a winch do I need for my car dolly?

I'd like to know what the tow dolly is being pulled with.

a 2000 lb. vehicle with good tires on a flat level surface should move with approx. 10% of it's weight implied in force, or, about 200 lbs. of force should get 2000 lbs. moving on a flat level paved surface.

As mentioned by many, the gradient resistance up onto the dolly is where the most effort will be needed.

Lifting a vehicle straight up takes more force than the actual weight of the vehicle.

When rigging wire rope through a snatch block or pully, you MUST be aware of your angles your wire rope is entering and exiting your snatch block/pully. It's at these points more force is implied on your wire rope. More often than not, catastrophic wire rope failures occur when angles are too steep and the working load limit/safety design of the wire rope used for the application aren't sufficient.

Anchorability is another piece to the puzzle. Figure a 66% "anchor factor" with brakes locked. This oughta keep you safe. Experience will teach you how far you can stretch this.