Rin and Yukio are fraternal twins born to Yuri Egin, as the result of being impregnated by Satan. Rin, however, is a girl while Yukio is a boy. To make things even more complicated, despite Yuri's last wishes being that Shirou Fujimoto take both children, Mephisto Pheles intervenes and, by way of a wager, takes Yukio to raise the child himself. Shirou is left with Rin, and takes her to the abbey in True Cross City as planned.
With some difficulty, Shirou raises the only female in the abbey to the best of his ability. But Rin grows on him quickly, and with a renewed sense of responsibility to some one else, he also opens up a small orphanage and takes in a handful of children by the time Rin is four.
When Rin is six, she gets in to a nasty fight at school, her hidden demon traits nearly breaking through and causing her to injure several students and a teacher. Fearing retribution from Shirou, she runs away and stumbles across a group of unruly teenagers smoking, drinking and gambling. They harassed her, pushing her around while they offered to "play" with her. Panicked and still running on the energy from the previous fight, she lashed out, and woke her flames for the first time.
Thinking it was some kind of cool magic trick, she decided to keep it under wraps and tried to bring it back out again, without much success. About a month after the first time, she was able to produce a tiny bit of flames again, but at the cost of utter exhaustion. So she kept working at it, again and again, eventually getting more and more but to the point of overtaxing herself until she passed out at school.
She backed off from practicing, letting Shirou believe she had simply come down with an illness, until at last she was able to do it well enough that she could finally show Shirou. He was shocked, and upset that her flames had woken so early, but he also resolved to give Rin proper training. Even if she could never become an exorcist in the certified sense, he wasn't about to leave her to her own devices.
He gave her a pendant that acted in a similar capacity to the scabbard of the Kurikara - the ancient demon slaying sword that had originally sealed Rin's power - by giving her an active seal that would only let her flames appear in an absolute emergency. He promised she would have the wards on it lifted bit by bit until she was completely capable of controlling them on her own. With this they began a regiment in a sealed room that hid her powers from the rest of the demonic world that involved martial arts katas and focus training with candles and other flammable materials. It was around this time that Rin took to the kitchen as another outlet, learning to cook by trial and error so that she could live off more than just barely passable curry and other plain foods.
When Rin was a little older, she got in to a fight with the older brother of one of the orphans she was friends with. As they argued she swore she could do things that Hisui could only dream of, and when he didn't believe her she removed her pendant to show him, and accidental burned Kohaku instead. Guilt wracked her little mind, and when she couldn't sleep she crawled in to Shirou's bed and confessed what she had done. Shirou, naturally, scolded her, but in the end the incident only strengthened her resolve to user her flames for good.
Eventually Shirou began taking her out on "moonlight missions" - not necessarily missions that took place at night but ones that were small enough that the Order wouldn't likely catch wind of what they were doing. She got hurt the first time out, causing Shirou to realize that she needed more than just her fists to carry her through. With that in mind he employed an old student of his - Shura Kirigakure. Shura begrudgingly agreed to teach Rin the sword.
Shirou didn't want to give her the Kurikara quite yet though, for fear of what completely unlocking Rin's powers would bring, but instead had a special sword crafted for her when she had become skilled enough with a training blade. Shura created a space of holding for her on a bracelet, and the weapon was concealed with in it, ready to be pulled at a moment's notice. Her training and the occasional missions with Shirou continued, though all the while Rin longed for something more. She felt soemthing was missing from her life, but she didn't know what it could be.
That was because all this time, Shirou had still been keeping a few things secret from Rin - namely that she had a brother, and who her real father was. Rin had, over time, come to understand that she was not fully human. Humans didn't burst in to flames, not even well trained exorcists. Humans couldn't hear other demons. Humans didn't have their teeth and ears grow and become pointed as they grew up. But as far as she understood she was likely the result of some poor human woman and a higher level demon that knocked her up for the hell of it.
Eventually Rin gave up on school all together. Even with the friends she had from the orphanage, she couldn't keep herself out of trouble, and she just wasn't skilled at all in keeping her grades anywhere above just below average. She went looking for work instead, since Shirou insisted she find some means to make a living for herself, but even that proved to be difficult.
While heading to an interview she was confronted by another girl she often had confrontations with - Rei Shiratori. Rei had a history of being cruel to people and animals alike, and some days before Rin had stopped her from tearing the legs off a pidgeon. She was back for round two with her cronies, and that's when everything went strange. Rei became posessed by one of the demon kings - Astaroth, the King of Rot, who throughout their fight dropped hints and clues that Rin was not mere the daughter of a demon.
Eventually Shirou managed to arrive and banish the demon from the girl, but Rin was left with more questions than answers. She demanded to know what was going on, and Shirou admitted the truth - her true father was none other than Satan himself.
Rin was furious to find such a thing out, and while she returned home and stewed about it long enough to be rightfully pissed at Shirou, she was still willing to listen to him because really... Shirou was the only father she ever knew. But the trust had been broken, and even though she decided in the end to forgive him because he did have his reasons, that didn't stop her from passive aggressively tormenting him for some time after.
Unfortunately the incident trickled through the right pipelines and Mephisto showed up some time later, demanding to know with that usual smile of his why Shirou had never told him that Rin's flames had awoken so long ago. Shirou did a good job of dancing around the answer, but their talk did eventually turn to the subject of Rin becoming an exorcist and attending the cram school at the prestigious True Cross Academy.
Shirou was adamantly against it, too afraid of what would happen if Rin's origins were revealed, but Mephisto put up a good enough argument for it - what better way to protect Satan's daughter than right under the noses of the very people set out to destroy him? Could Shirou really protect her forever on his own, knowing that sooner or later she would have to make her way in the real world without him at her back every step of the way?
Eventually Shirou agreed, and to add salt to the wound, was forced to reveal that Rin also had a brother who was a cram school student. Naturally Rin was upset at yet another secret being told to her, but there was little she could do about it aside from fume, and quietly dread what was going to come of that first meeting...