Biography
Dr. Elizabeth Klein was born in an alternative timeline, one in which Nazi Germany won the second World War. She was a favourite of Adolf Hitler and assigned many responsibilities, many of which were related to unknown phenomena which posed a threat to the Reich. One such assignment was to investigate the outbreak of a seemingly airborne viral agent in occupied Kenya after the Reich crushed the Mau-Mau uprising in their reality. She was unsuccessful in solving this mystery, and it continued to haunt her throughout her life. Later, she was tasked with examining a strange blue box found in Colditz Castle in 1945. Aiding her in this assignment was the mysterious Johann Schmidt, who was, apparently, an alternate version of the Doctor's eighth incarnation. Schmidt tricked her into piloting the TARDIS back to Colditz in 1945, knowing that his past self would be able to use it to correct the timeline. The Doctor did so, but stranded Klein in the traditional universe in the process. Furious at the Doctor's trickery, Klein vowed to one day restore her world, which she believed to be the "correct" version of reality.
Along with other Nazis who fled the Allied invasion of Europe and the fall of Berlin, Klein ended up in South America, where she became associated with Hans De Flores. There she got a medical doctorate in addition to her doctorate in physics. Afterwards, she set out for Africa, where she hoped that the events of her timeline would repeat themselves in the 1950's of our universe, thus giving her another chance to discover the source of the mysterious virus. Isolated in a house with several other women during the Mau-Mau rebellion, she once again met the Doctor. Despite the animosity between them, they worked together to solve the mystery of the virus and stop an alien plot to use Earth as a testing ground for biological warfare. The Doctor, knowing that Klein would continue to pose a threat to history, insisted that she accompany him as a companion on his travels so that he could keep an eye on her. (BFA: A Thousand Tiny Wings)
Personality
Klein is a fervent believer in National Socialism, and the benefits to society of fascism in general. Despite holding the view that the needs of the many outweigh the needs of the few, she does form close, compassionate relationships with the people under her care, as evidenced by the impact Lucy's death had on her (A Thousand Tiny Wings). Despite knowing the atrocities which Nazis like Mengele committed, she draws a differentiation between their actions and their ideals, saying that Mengele went too far. However, she is not beyond coolly executing an enemy or torturing them for information should that be necessary. She is not intimidated by the Doctor or his reputation. She is not a hypocrite, in so far as she is willing to die in order for the greater number of people to survive, although she does admit that she would prefer not to die.