The doorjamb (where the latch meets the door frame) needs to be insulated from sound on the face of the door. Start by adding rubber weather stripping over the space between the door and frame. This will reduce (but not eliminate) any sound. Run the rubber stripping around the face of the door and across the bottom. Use a light source behind the door to check for light leaks.
The Jamb Plate
The jamb plate (a piece of metal used to reinforce the door frame where the latch holds the door closed) needs insulation to reduce the metal-on-metal contact sound. A quick and easy fix is add a few layers of duct or masking tape to the plate. Be sure, though, not to make it so thick it hinders the latch's operation. Another more permanent solution involves Plasti-dip liquid rubber. Plasti-dip is sold in home supply stores for adding rubber handles to metal tools. Use a small paintbrush to apply several coats of the liquid to the metal plate and allow each coat to dry thoroughly. Now the door latch will make contact with the tape or rubber and won't make a sound.
Add weather stripping inside the door to act as a seal not only against air but also sound. Be sure to take care getting the seal tight in the corners. If possible, close the door and use a bright light to look for leaks. When applying the seal take extra care in these areas. Again, consider using tape to build up areas of the door frame before applying the weather stripping. This will increase the thickness of the stripping and make a tighter seal.
Unfortunately, when turning the door knob mechanism there is very little sound proofing options. The best thing to do is make sure that the mechanism is well lubed with dry graphite and produces as little sound as possible. Although comparatively expensive, there are doorknob mechanisms made of nylon and other hard plastics. These are not as durable as metal but much quieter. Look for them from a music studio or laboratory supply company.