To dream of biting someone could be a warning that you are overdoing the pressure you are applying to certain people and causing them pain. In other instances it tells of low business practices and a lack of ethics in all your dealings.
If you are biting someone as a vampire then you feel like you must make your own way and not be dependent on others, which makes you feel like a ‘bloodsucker’. See Vampire.... Encyclopedia of Dreams
Emotional hunger, experiencing childhood longings, dependence or feelings. Being sucked at: if other than child-parent relationship, feeling one s energy drained; having a sense of another person, or an aspect of oneself, being parasitical—not giving as well as receiving—or having baby needs. Idioms: suck up to someone; bloodsucker. ... A Guide to Dreams and Sleep Experiences