Isn't it obvious? All I have to do is beat the crap out of her.

When first introduced to Kyouko, we see a very selfish girl -- and she is. Because they don't produce grief seeds, she stops Sayaka from destroying the familiar (a magical being that will eventually evolve into a witch) she had been pursuing in the alleyway. If killing it serves no benefit to her, why bother? We also see her sadistic side very early on: she explains to Sayaka that -- like other organisms -- there is a food chain when it comes to being a magical girl. "The witches eat the weak humans. And we eat those witches." In a later episode, confronting Sayaka outside of Kyousuke's home, she suggests that Sayaka should use her magic to cripple him so that he has no choice but to rely on her. When Sayaka begins to grow furious, Kyouko offers to do the deed herself. She sounds pretty unlikable at this point, right? But - of course - there is more than meets the eye. Kyouko's selfishness comes with good reason.

The thing I wished for destroyed my whole family. I brought all this suffering down on my family 'cause I made a wish for my dad without knowing what he really wanted.

Born into a religious family, Kyouko lived with her mother, her younger sister - Momo - and her father, who was a preacher. Her father was a very kind man, who Kyouko says would cry while reading the newspaper because of the state of the world was in. Believing things needed to change, he began to preach ideas that strayed from the church's accepted and recognized teachings. The churchgoers did not support his methods and eventually stopped attending her father's sermons. He was excommunicated from the church, and Kyouko's family sank to rock bottom. With no one to believe in him besides his family, there was no way for him to support them; they lost their home and weren't able to have any of the bear necessities, like food. Kyouko, distressed that no one would give her father the time of day, was unable to accept their situation. After being contacted by Kyubey, she wished for everyone to listen to and believe in her father's words, thus becoming a magical girl. Things immediately started to look up. The day after, people from all over flocked to her father's church in unbelievable amounts. While her father shared the word of God he believed in, Kyouko fought witches on the sidelines, proud of herself and what she was now able to do. However, the happiness her family found didn't last long. Eventually her father found out the real reason for the sudden influx of followers and quickly descended into madness. He called Kyouko a "deceiving witch" and started drinking excessively, his breakdown leading him to murder his family and his own suicide. Only Kyouko survived. This event changes Kyouko drastically. By the time Kyouko appears in the anime, she has already been a magical girl for ~2 years. Since then, she's subconsciously abandoned her old powers for the new magic she's learned. Selfish as she may be, this is evidence that she can feel guilt. It's easy to make the connection that she's ignored her wish-given powers because they are ultimately what caused the death of her entire family.

You don't get it when I explain it, and trying to beat some sense into you ain't really working. Guess my only option is to kill you!

Kyouko is quick to jump to violence in situations, especially in those where she deems people unable to listen to her reasoning as idiots and decides that there is no other way to get through to them. This doesn't seem to bother her much; when Kyouko fights, she does it with a grin on her face. She shows no qualms about killing Sayaka whatsoever. With Mami dead, she considers Mitakihara territory her own, and the only thing that stops her from her goal is Homura. The way she fights can also be linked back to the sadistic side she sometimes shows. She'll egg people on (as she does in the earlier episodes with Sayaka) to get them wound up enough to fight, and once challenged, Kyouko does not like to be interrupted or have things cut short.

If I see you wasting food like that again, I will kill you.

Being a person who does a lot of things on her own, Kyouko is a bit socially inept. Whether she does it to protect herself or not, Kyouko does not attempt to form any bonds with the other girls and refers to them only as "you" (in Japanese she uses the word "anta" instead of "anata" or "kimi." "anta" which is an impolite way of referring to someone in Japan.) and doesn't hesitate to throw swear words and insults around. It isn't until she connects with Sayaka that she calls her by name, the only other one to be formally addressed by her being Mami. On the other hand, if she approaches someone looking for a truce or a friendship or to be comrades in arms, Kyouko will "seal the deal" with an offering of whatever food she might be eating at the time. Because of how long her family went without it, Kyouko treasures food as if it were gold. If someone wastes food in front of her, such as Sayaka did in the church, she impulsively reacts and will not hesitate to "punish" them, from a punch or even strangling someone. Giving away her food to someone else - no matter how much she has - is a big deal.

Stories where love and courage win out in the end always turn out like that, you know?

While we see a lot of her selfish and mean-spirited attitude, Kyouko doesn't lack the ability to sympathize or care about people other than herself. Once recognizing that Sayaka is walking on the same type of path she had before everything fell apart, she starts to look out for her. She sees herself in Sayaka and thus wants to prevent her from getting hurt whether it be mentally or emotionally, even when she denies Kyouko's request for her to change her ways. Kyouko shows up numerous times to try and protect Sayaka; while fighting Elsa Maria, when Homura attempts to kill Sayaka before she meets her inevitable fate, searching for her until she finds her in the train station and retrieving her body from Oktavia's lair until she is able to find a solution to (what ends up being) her short-lived quest. Like Sayaka, Kyouko was very much an idealist. She believed in fairy tale endings, the kind where justice prevails and the good guys end up happy while the baddies get stomped on. After her family falls to shambles, Kyouko's idealistic views crumble and slip through her fingers and she makes the choice to use magic only for herself. Sayaka, however, was able to reawaken her to the feelings she once felt, and that's part of what drives Kyouko to try and save her.

As long as you protect the one you want, to protect them to the very end is enough. And here I thought that's what I was doing until now...

Warding off Oktavia's attacks, she recalls her first meeting with Sayaka and how they've acted around each other since. She reveals that she admires Sayaka's stubbornness in a way; no matter how much Kyouko would beat her up or call her things like naive, Sayaka would stand right back up again. Kyouko had once been like that too: a person who believed in stories where loved and courage prevailed with a fairy tale ending. She tells her that she understands her anger but wants her to wake up and use her powers like she wanted to. Finally understanding and accepting that there is no other way around it, Kyouko resolves to kill Sayaka as an act of mercy. She not only shows concern for Sayaka but also Homura and Madoka, advising Homura to take Madoka away from Oktavia's lair, having already put Madoka in enough danger by taking her along. Summoning a gigantic version of her spear, she reassures Sayaka that she doesn't have to worry. Kyouko will stay with her and not let her die alone, she won't let her be all by herself. She knows exactly how it feels to be alone. The attack is so powerful that it shatters Kyouko's soul gem, and with it, takes her own life and Sayaka's. For someone so determined to use magic only for herself, she ends her life with a very selfless act.