Kaname is Kinji's younger sister, and with that comes a fierce loyalty of a family. She is willing to block bullets with her body without second thought just to protect him.
However, it is implied that she has a bit of a brother complex for him as she ends up taking Kinji to the movies, has him buy her things, and all sorts of things normally only couples would do. As a result, she is very protective of him in that regard, however, she mentioned that brothers and sisters shouldn't have this kind of relationship.
She is extremely serious about keeping Kinji away from other girls, and is willing to kill the other members of Team Baskerville just to make her point. After the dual setback of her HSS backfiring on her and a loss in a Lumberjack duel against Shirayuki and Jeanne, Kaname has become more of a normal girl and no longer threatens to kill others. However she still tries to separate her brother from the other girls and desire's his attention.
While Kinji was attending normal school Kaname and Kinzou (GIII) would visit their grandparents house where he was residing to be with him. Also, she would be maintaining his Beretta for him.