He became the bellringer twenty years ago when the film's villain, Judge Claude Frollo murdered his mother. He came closing to killing Quasimodo by throwing him in a well. Notre Dame's archdeacon then told the evil judge that for killing Quasimodo's mother, he was doomed to damnation unless he raised the child. Naming the child Quasimodo, Frollo made him live in the bell tower.
Despite having immense strength, Quasimodo avoids violence and is extremely pure hearted (despite being raised by Frollo), and even tried to save Frollo even after learning Frollo killed his mother. This can allow him to be overpowered and captured.
He originally fell in love with Esmeralda but became broken hearted when she fell for Phoebus. At the end of the movie he gets over it and gives the two his blessing.
He finds love again in the sequel in the form of Madellaine and ends with the two getting together.