When Kim was just four years old her parents went on a vacation but never returned; they were killed by the dolem Ritardando. Fourteen years later she swore an oath of revenge and decides to work with Terra and kill anything that's part of the Mulian world, especially the dolems. When Ritardando appeared again early in the series, Kim became very emotional and lashed out her frustrations on the then reluctant Ayato. Because Ayato had such a unique gift to pilot the RahXephon, she expected more of him. It wasn't until after Ayato fought the Dolem that the two made peace.
Kim holds a secret relationship with Souichi Yagumo mostly because she's very concerned of Megumi's feelings. Her and Megumi have been best friends since they started working together, and Kim is very well aware of Megumi's feelings towards Yagumo, so she decides to keep their relationship a secret. It wasn't until Megumi found out about their relationship by accident that Kim finally let it all out and started crying because she didn't have the strength to tell Megumi about her relationship with Yagumo. It is also strongly hinted that Kim that is pregnant about midway through the series.
When Yagumo is killed by Rahxephon-Ayato towards the end of the series, Kim once again lashes out at Ayato, though this time the hatred is (obviously) much stronger. She even attempts to kill Ayato by shooting him but Haruka tries to prevent it. She does land a shot, but by this point Ayato had become one with the RahXephon and doesn't take any damage. Frustrated, Kim sort of gives up on everything. However, she could very well be very happy with Yagumo in the world after the tuning.