Picard: Entered Starfleet against his fathers' wishes. He receives a "dressing down" from his father when they meet in the afterlife in episode 141 "Tapestry". Picard also had a very antagonistic relationship with his brother.
Will Riker: His mother died when he was very young and had a very strained relationship with his father afterwards. Riker left home when he was 12 yrs old per episode 40 "The Icarus Factor".
Worf: He lost both parents in the Battle of Khitomer. Adopted by a human couple on Earth.
Dr. Crusher: Wasn't she was raised by her grandmother per episode 166,"Sub Rosa"? Don't know what happened to her parents. And, of course, she lost her husband to an accident, leaving Wesley Crusher without a father at a very young age.
Deanna Troi: Her father died (accident or illness?) when she was very young. Her mother, Lwaxana Troi is very overbearing.
Data: Noonian Soong abandoned his "son" on Kiron III as he fled from the crystalline entity.
Tasha Yar: Had a violent childhood as an orphan.
I think the only officer who had a somewhat normal childhood was Geordi LaForge.