Snape became a teacher to protect Harry and as penance for Lily's death
Snape felt terribly guilty of his own role in Lily's death. Dumbledore made use of that guilt, and made him promise to protect Harry Potter. Therefore he became a teacher at Hogwarts, because he had to be close to Harry in order to protect him.
“If you loved Lily Evans, if you truly loved her, then your way forward is clear.” Snape seemed to peer through a haze of pain, and Dumbledore’s words appeared to take a long time to reach him.
“You know how and why she died. Make sure it was not in vain. Help me protect Lily’s son.”
There was a long pause, and slowly Snape regained control of himself, mastered his own breathing. At last he said, “Very well. Very well. But never – never tell, Dumbledore! This must be between us! Swear it! I cannot bear…especially Potter’s son…I want your word!”
-Harry Potter and the Deathly Hallows
Since Voldemort was ultimately responsible for Lily's death, Snape joined the Order of the Phoenix, and in order to be active in the cause against the Dark Lord, he needed a credible excuse to be in contact with the headmaster. Teaching proved to be a good cover, since Voldemort never realized Snape's betrayal until his death.