Doing something bitterly means doing it in an angry, resentful way. If you argue bitterly with your best friend, you're both expressing hurt feelings and hostility — it may take a while to mend your friendship.
When people speak bitterly to each other, there are a lot of heated emotions involved. If you are bitterly opposed to the dress code at your school, you're truly angry about not being allowed to wear your favorite T-shirt. Bitterly also means "harshly or bitingly," as when the wind blows bitterly in your face as you walk along the beach in February. This adverb shares a root with bite.