Is it okay to have blood in your eyes if you have a black eye?

Depends on the cause: If the eye is hit hard enough to bruise the eyelids (causing a black eye), it could be hard enough to cause the surface vessels of the eyeball to break. Any hit hard enough to do this should be evaluated by an eye doctor.

Sub conj hemorrhage: With a black eye, blood can certainly spread to the white of the eye, but could be indicative of a more serious injury or fracture. If you are describing blood inside the chamber of the eye then it is most serious and should be evaluated.