Broussard was born in Marksville, the seat of Avoyelles Parish in south central Louisiana. He moved to Kenner, a city in Jefferson Parish, in 1958. He received a law degree from Loyola University New Orleans in 1973. The following year he was elected to the Jefferson Parish School Board and reelected in 1976. In 1977, he was elected as a district chairman to the Jefferson Parish Council and re-elected in 1981. He successfully ran for mayor of Kenner in 1982, and he was reelected in 1986, 1990, and 1994.