1-Remove the insert.
2-Be sure that the Zippo is empty. If you are not sure, leave the insert out in a ventilated room for 1 or 2 days.
3-Screw off the flint holder, take it out carefully, along with the flint.
4-Remove the cotton flap, then, with needle nose pliers, carefully remove the "cotton" balls inside of the Zippo. There should be 5 of them. Keep them nearby.
5- If you have not yet changed your wick, then simply pull up on the wick and snip off the black part. Pull the wick back carefully so that it won't go out of the Zippo. The wick should go almost all the way to the top of the Zippo "chimney".

(-To replace a wick, simply pull out the left over wick, and replace with a Zippo wick. Then follow step 6.)
Tip: What I find that helps a lot to keep the wick from burning black again, is I fill up the Zippo completely, then I apply a few drops onto the wick, and let everything soak into the wick for about 15 minutes.