Her grandmother on her mother’s side was born in colonial India in the Lucknow province, which gives Clarke one-eighth Indian ancestry.
Clarke was born on 23 October 1986 in Westminster, London and grew up in Berkshire and Oxfordshire. Her father was a theatre sound engineer from Wolverhampton, West Midlands. Her mother is the vice-president of marketing for a global management consultancy. She has an older brother.
Clarke is of partial Indian descent as she stated in a 2018 interview that her maternal grandmother was born as the result of a secret affair between her great-grandmother and a man from the Indian subcontinent. Her grandmother wore light makeup to hide the dark complexion she had inherited from her father. She credits this background for her family having a "history of fighters", saying: "The fact that (my grandmother) had to hide her skin color, essentially, and try desperately to fit in with everyone else must've been incredibly difficult."