How to clean you clarinet according to wikihow.com
Remove the reed after playing and store it in a reed holder to dry. Put the rest of the clarinet in a safe place that is not too hot or to cold. Don't stand it on end, unless you have a clarinet stand (it holds your clarinet upright, and can usually be folded and stored in the bell), since it can fall over.
Use your polishing cloth to wipe fingerprints off the ligature and keywork. Then, carefully place the cloth back in your case. Don't use any kind of metal polish (unless the clarinet is all metal, ie: silver) or other substance on your instrument, except for key oil. Key oil makes the keys move more easily.
Use your swab (made of silk, nothing else, other materials can damage the insides) and drop the cloth's weight and string through the clarinet from the bell to the mouthpiece and pull it through. You may have to do this several times. Take the pieces apart and dry the tenons (these are the places where the joints of the horn fit together.) Then, let the swab dry before putting it in the case, otherwise your keys will be discolored.
Get out some cork grease. Grease the corks by rubbing cork grease onthem with your fingers. Cork grease comes in many different forms, but the most common looks like lip balm. However, lip balm is not an appropriate substitute for cork grease, and cork grease can be poisonous if ingested, so do not confuse the two. You don't need to do this EACH time you take your clarinet apart, or even very often, as too much cork grease will weaken the corks and make them prone to breaking apart. Just examine your corks regularly and apply cork grease as needed depending upon the dryness or wear in the cork. You may need to grease them more often in the dry winter months. If the instrument is difficult to put together or take apart put cork grease on the corks.
Take out your cleaning cloth and polish the keys to remove fingerprints and grime. Then you may, if necessary, take out your dusting stick. Dusting sticks are long, twisted, thin, wire, with a small, broom-like thing at one end, and a little thing that looks like a tooth cleaner at the other end. Gently clean between and under the keys with the mouthpiece-brush end, and brush the dust off with the broom end. Pay close attention to the tone holes that your fingers cover as they often collect gunk and grime from your fingers. You may wish to polish the instrument again after doing this. This would also be a good time to oil the keys, if necessary.
You should also make sure to swab the insides of the clarinets joints, with or without cork, to get rid of the excess saliva there.
Periodically use a q-tip and wipe it around the inside and outside of the mouthpiece to remove the gunk that will collect.
Place the mouthpiece cap over the mouthpiece, then store the clarinet in its case. Put the case in a safe place.