I have a boil on my knee. What is the quickest way to heal it?

Warm compress & Tylenol (acetaminophen). You may get by with letting it pop, aided by warm compress, and relieve the pain with tylenol (acetaminophen) or Motrin and the like. If that does not relieve the boil in a couple of days, you should see your doctor to see if needs to be drained.
Boil=Abscess. A boil occurs when there is an infection of one of the many skin glands. If there is simply infection, then antibiotics will do the trick. However, most boils are collections of pus (abscess) which means that even if the infection gets better, the pus has to be absorbed which takes time. Opening abscess releases pus which promotes faster healing. Simple dressings to the wound change 3 x/week.