She's depicted as a genius who tries to acquire national influence and power despite being a theoretically powerless princess. I'd say her relationship with Climb is a sort of isolated, miniature test case of her machiavellian view of life. She saved Climb from death and absolute powerty and is maintaining a very carefully designed distance to him, creating an everlasting and bottomless loyalty. Climb isn't especially capable, which has been repeated to death, but that's an essential part of the equation, since if he was more talented, he wouldn't need the princess as much. I'm not even sure if she's a yandere at all. If she is, it's probably only a second nature of hers. Climb is a dear tool to her.
MFauli, what you are seeing here is the plot of the story.