Quoted from "Filmtracks":

Elfman spent his childhood in Los Angeles. When he turned 18, he moved with his brother, Richard, to France. There, he joined a theater group and became acquainted with the many different instruments of the orchestra. After spending a few years in France, he moved to Africa. He moved back to the U.S. after contracting malaria. He still considers his time in Africa a great benefit to himself.

In 1980, Elfman worked with his brother, director Richard Elfman, on the film Forbidden Zone. He formed a group known as "The Mystic Knights of Oingo Boingo." They performed the score that Elfman wrote for Forbidden Zone. Oingo Boingo, as it became known, blossomed into a permanent group and drew a large cult following (in 1995, however, they broke up).

In 1985 Elfman met Tim Burton, who asked Elfman to score the music for Pee-Wee’s Big Adventure. This was Tim’s first time directing and Elfman’s first time composing for a big-budget film. Elfman called in his friend and fellow Oingo Boingo member, Steve Bartek, to orchestrate for him and Pee-Wee was a success. Thus began the friendship between Elfman and Burton. Elfman first became known by soundtrack fans in general with his funky, light-hearted score for Beetlejuice. Combined with Harry Belafonte’s songs, the album was an enormous cult success.

Today, they are still working together as a team. Elfman has scored the music for every single movie that Tim has directed, with the exception of Ed Wood. He won his only Grammy in 1989 for his theme to Batman. Elfman considers Batman to be one of his best scores, and believes that it should receive far more recognition. He has never been nominated for an academy award. Elfman, along with his fans, believe that the Academy doesn’t like him because of the fact that he is completely self-taught. He keeps his distance from the rest of the composer community, which he considers "elitist."

- http://www.filmtracks.com/composers/elfman_bio.html