Staple mass-loaded vinyl to the drywall of the ceiling below the upstairs floor you want to soundproof. This material has more density than lead and will absorb vibrations as well as sound from the floor above.
Screw wooden furring strips through the mass-loaded vinyl and into the drywall of the ceiling. Arrange the furring strips so that they are approximately 24 inches from one another and running perpendicular to the joists in the ceiling. Doing so will create a disconnect between the original ceiling and the new ceiling.
Cover these furring strips with a new layer of drywall to create a new ceiling. Attach this drywall to the furring strips with drywall screws.
Seal the drywall seams with acoustic sealant to keep sound from coming through the cracks.