Since Then: Christian took her with him to Australia, allegedly as a bodyguard. He eventually instead asked her to drink and "be pathetic" with him. She rejected this invitation and also had to physically restrain him when he visited Lindsey's house, demanding to see Claire. She never learned his real name.
Since Then: The fling produced a child, and Christian repeatedly visited Australia to visit her. Eventually, Carole stopped allowing these visits. When Carole got into a car accident, Christian paid the medical bills. She also tried to get their daughter to turn off the life support. Carole recovered though, and years later ended up attending Christian's funeral.
Origin: Christian fathered Claire after a fling with her mother.
Since Then: Christian visited Claire regularly when she was young, reading and singing to her. Her mother later discouraged his visits. Christian returned to Australia years later to pay Claire's mother's hospital bills, and he finally revealed himself to his daughter. He encouraged her to let her mother die. On the island, the Man in Black first contacted her using Christian's form.
Origin: Christian told Jack as a child not to try and fix everything all the time because he wouldnt have what it takes when he failed to do so.
Prize: Control over Jack's destiny. Christian is fighting to regain Jack's respect, Jack is fighting to free himself of his father's influence.
Fuel: Christian pressured Jack to sign a report that a patient Christian had operated while drunk arrived beyond hope of recovery. Upon finding out that the patient had been pregnant, Jack reversed his statement and turned Christian in. Christian lost his license and continued to struggle with alcohol ("All the Best Cowboys Have Daddy Issues"). Jack's wife, Sarah, sponsored Christian into Alcoholics Anonymous and Jack misinterpreted this as an inappropriate relationship between his wife and his father. He followed his father to an AA meeting where he punched him out ("A Tale of Two Cities"). Christian caught a flight to Australia and went on a bender ("Outlaws"). Jack's mother sent him to Australia to retrieve Christian ("White Rabbit").
Resolved: Yes. While Christian died with many of their issues unresolved, in the flash-sideways timeline they forgive each other instantly since they are both happy just to be reunited once more. They each tell each other how much they love each other and Christian helps Jack let go. ("The End")