Does pink eye cause pain in the back of the eye?

Not usually. Pink eye pain is usually more surface irritation. There could be a sinus pain associated with pink eye or it could be something more serious that needs evaluating.
It can. When any body part becomes inflamed and irritated, referred pain can be felt in surrounding tissues. The eye has a lot of pain receptors and irritation from severe conjunctivitis can radiate to an appparent location behind the eye.