Complex sentence: His first two terms as chief minister came to be when he replaced Jayalalithaa, who was forced to resign by the courts; his third and current term began upon Chief Minister Jayalalithaa's death.
Split into simpler sentences: His first two terms as chief minister came to be when he replaced Jayalalithaa in the role, after she was forced to resign by the courts. His third and current term began following Chief Minister Jayalalithaa's death.