Born and raised in Darjeeling, India, Avari was born into an old Parsi-Zoroastrian family, and his father, Eruch Avari ran two movie theatres in Darjeeling. Along with this, Avari's father was once the Chief Game Warden of the West Bengal Forests. Avari's early education was as day scholar at the well-known North Point School, and Avari studied at the College of Charleston in South Carolina. A member of the Avari-Madan family of Calcutta and Darjeeling, Avari's great-grandfather was Sir J.J. Madan.
Avari has portrayed over two dozen different nationalities in his career, and has made appearances onstage as well as screen. Avari is married to Margaret Walker.