Use a staple gun to fasten pieces of mass-loaded vinyl to the existing wall. Cover the entire wall with this material, overlapping the edges.
Seal the overlapping edges of the mass-loaded vinyl with an acoustic sealant. This prevents sound from leaking through the seams.
Screw wooden furring strips into the drywall behind the mass-loaded vinyl. Align the furring strips so they are perpendicular to the studs in the wall; doing this creates a disconnect between the original drywall and the new drywall you will attach to these furring strips. This helps make the final layer of drywall more sound resistant.
Screw new pieces of drywall into these furring strips, creating a finished look to the modified wall.