When Kodocha starts, it seems Hayama's older sister Natsumi has to take care of the house and her brother all by herself. Her mother died giving birth to Hayama, and her dad's time and attention is occupied with work, so she harbors a strong resentment towards Hayama. She blames him for their mother's death and constantly calls him a "child of the devil." She even refuses to make dinner for him, so Hayama has to eat outside alone…
As much as she's easy to dislike for her actions, Natsumi's not really that hard to understand. Her attitude is extreme and selfish, but it comes from the fact that she's basically just a kid with no one to care for her. She pretty much lost both her parents and her (presumably) happy existence when Hayama came into her life. Add on top of that the fact that she was very young when it happened, and it's not that surprising that she developed an irrational hatred for her brother.
It's clear that as she grew older, Natsumi had to take on adult responsibilities for her brother's sake, so in her eyes he is the reason she can't have a happy, normal life (and the reason her dad is such a distant person).
In spite of this, Natsumi eventually realizes her mistakes thanks to Sana's TV drama. In it, she has the chance to actually see her situation and her actions from a more objective point of view, or at least from the outside. She sees herself in Asako's character and finally realizes the extent of pain she's caused her brother. Also, for the first time she actually stops to think about the fact that her mom's death wasn't Hayama's fault, and that her mom wanted to give him life.
After that experience (and once her dad begins to be there for his family) Natsumi changes drastically and becomes a fairly ordinary older sister. She teases Hayama sometimes, encourages him to do things he should (like thank Sana for sending him an invitation to her play), tries to boss him around once in a while, and I think she really looks after him. Even though she can be bit crazy sometimes, overall Natsumi's a pretty nice older sister.
