Kali was a Kaliratha, a member of a race of four-dimensional beings. She had four arms and possessed four swords, through which she could channel her four-dimensional powers and slice through the fabric of time and space.
As was common for Kaliratha, when she came to Earth, to India, she presented herself as a deity. (COMIC: The Swords of Kali) She was worshipped as the Hindu goddess of death and destruction (PROSE: Endgame) and also inspired the religion of Thuggee, who dedicated their victims to her. She fed upon the life-forces of her followers, but rewarded her most loyal disciples, the Scindia family. She laced their genetic structures with her own DNA granting them immortality.
After the Hyperion War, Kaliratha were deemed one of the malevolent species and were wiped or incarcerated by the Alliance of Races. Kali put up an enormous fight laying waste to the surroundings and recorded in scrolls by mankind. But ultimately she was destroyed, her body was incinerated, and her four swords were secreted into hidden temples protected by killer androids. Since that day the Scindia family worked tireless towards her resurrection.
Tiger Maratha retrieved three of her four swords for them. In 2315, Chandra Scindia coerced the Twelfth Doctor into getting the last sword. And he released the souls of the dead into a chamber with Clara Oswald causing Kali to be resurrected in Clara's body, which undergone dramatic changes. Clara-turned-Kali substantially grew in size, had four arms and blue skin, as well as eyes glowing green.