Remove the insulation from the existing ceiling. The neatest way to do this is to go into the attic and bag it up. The easiest (and messiest) way to do it is to knock the ceiling out with a sledgehammer and crowbar and let the insulation fall down along with the ceiling. If you're not sure what the insulation is made of, have it tested before you remove it. If it has asbestos in it, hire professionals to do the job safely. In any case, wear coveralls, gloves and a dust mask to minimize your exposure.
Clean up the remains of the ceiling and insulation to maintain a neat and safe work area.
Assess the joists that were supporting the ceiling, calling in professional help, such as a structural engineer, if necessary. These joists not only support the ceiling -- they also prevent the roof eaves from splaying. If you remove them to create maximum exposure for the cathedral ceiling, you will need to accommodate their loss with cross beams or exterior buttresses to reinforce the exterior walls.
Insulate between the roof rafters. Use 6-inch-thick fiberglass or rockwool batt insulation. Don't press the insulation up against the underside of the roof. There needs to be an air space between the top of the insulation and the underside of the roof sheathing to allow ventilation of the roof.
Attach drywall to the underside of the rafters using 1 1/2-inch-long drywall screws. Begin at the eaves, trim the sheets of drywall as necessary to fit the ceiling spaces and work up toward the peak. Having several assistants makes this job much easier.
Apply spackle to the joints between the drywall sheets and to the holes made by the screws. Apply drywall tape to the joints, put more spackle on top of it and feather it out with the drywall knife to create a smooth surface. Allow it to dry for 24 hours.
Sand any irregularities out of the surface of the spackle. Paint the new ceiling with primer and then with two coats of ceiling paint.