She is normally a rather innocent girl, but has no problem being rude or brash to certain people. Her her father she shows nothing but kindness and loyalty, and receive the same in return.
She is very close with the Marine captain Smoker. Close enough to where she calls him Uncle Smokey.
It's unknown what fate befell Carol's mother.
She first appeared in the One Piece anime during the Loguetown Arc in the filler arcs that took place in the middle of the canon arc. The Straw Hat Pirate Usopp was trying to purchase a special set of North Blue Goggles, but forgot his wallet outside with his backpack. While he was distracted, Carol stepped in and bought the only pair of goggles as a birthday present for her father. She left and mocked Usopp as an old man and left.
Usopp chased after Carol and tried to intimidate her into giving him the goggles, but she didn't believe him. He went so far as to lie that he had a 30 Million Berries bounty on his head. Though she was slightly impressed. Carol's father had then appeared behind them. She rushed to her fathers side to give him the present. She also accused Usopp of harassing her and trying to take the present. She even told her bounty hunting father about the bounty Usopp claimed to have.
Daddy went through the latest wanted posters he had picked up and did find Luffy's. Usopp tried to escape, but Daddy was sharp eyed enough to spot the back of Usopp's head in the background of the photo on Luffy's wanted poster.
Carol, same as her father, holds a special place in the One Piece series. The series creator Eiichiro Oda had wanted to add the story of Daddy and Carol to explain there Usopp got his goggles. Oda was faced with pressure to get the story moving faster and get the crew to the Grand Line. Though they never appeared in the official manga story. This tale was told in the anime filler arcs about Loguetown.