FAMILY
It would be accurate to describe Asuka as having come from a broken home. Her father, Langley (first name unknown), is an American doctor and her mother, Sohryu Kyoko Zeppelin, was half German, half Japanese. What their marriage was like before Kyoko engaged in the contact experiment with EVA-02 is unknown but after the experiment, it certainly wasn't a pleasant one.
Father
When Kyoko was admitted into the hospital for her severe mental health issues, Langley began an affair with Kyoko's doctor. He was completely indiscrete about his relationship with her, going as far as engaging in sexual activity in the room adjacent to Kyoko's. He feels that Kyoko spent too much time with her research, largely ignoring their daughter. By the time Kyoko's funeral comes to pass, he's already remarried.
Stepmother
As Langley's new wife, Kyoko's doctor is now Asuka's step-mother; she clearly had no reservations about having an affair with her patient's husband. We hear her telling Langely that she doesn't really like Asuka, that her independent attitude creeps her out. In episode 22, we see that she calls Asuka periodically to catch up but Asuka explains to Shinji that these phone calls are for keeping up appearances; she doesn't really get along well with her stepmother.
Mother
As an employee of Project E in the Germany branch, Kyoko spent much of her time being involved in her research. She performed a contact experiment with EVA-02 which resulted in only part of her soul -- her motherly qualities -- being absorbed into the Evangelion. With an incomplete soul, Kyoko lost her mind and was admitted to a hospital for care.Kyoko became obsessed with rag dolls, caring for them as if they were actually her daughter while the real Asuka watched on. Kyoko would switch between being overly caring and horribly violent toward the dolls, ripping the head off one and strangling another. This did not go unnoticed by Asuka. Kyoko would talk to the dolls and refer to Asuka as "that girl over there" and when she eventually committed suicide, she also hung the doll along with her. She wanted Asuka to die with her, as she believed neither of them were wanted by her husband and no longer had any purpose.
Asuka did and still does love her mother very much. She wanted to impress Kyoko, hoping that her selection as an Evangelion pilot would free them both from her father. Sadly, little did Asuka know that her selection as the Second Children was a direct result from Kyoko's contact experiment and suicide; a pilot can only synchronize with an Evangelion unit that has the soul of their mother within it.