First things first, stable your current pet,
then go find a tamable creature with bite rank 4 and tame it,
fight with this new pet until YOU have learned bite rank 4 (it'll tell you in the chat frame),
ditch the new pet, pull your old pet out of the stable and teach it the new skill!
Fizzwidget's Hunters Helper is incredibly usefull for this, as it shows you what abilities you can learn from a creature in it's tooltip.