Hermione Granger was many things, but technically a Muggle-born witch wasn't one of them. While the names that came with the being a Muggle-born weren't exactly pleasant and the prejudice against her was less than ideal, she still immensely preferred the blood slurs over what would happen to her if they knew the truth. It hid her well—no one would delve too deeply into her past if they believed she was born to two Muggle parents.
The plan was her mother's idea. She had perfected it during one of her moments of lucidity which had occurred much more often while Hermione was still very young. Her mother's magic had become incredibly weak, but she found the strength to alter the minds of the isolated Granger family and impart upon them her own daughter for them to take in as their own. Illegal and immoral, yes, but Hermione's mother considered it a small evil in the grand scheme of things. It was nothing when it came to protecting herself and her daughter. However, incapable of giving her up completely, Hermione came to know two very different sets of parents as she grew older.
Hermione loved the Grangers. With the practicality that came from being dentists, she enjoyed the structure and security they provided. They fostered her love for books as Hermione's thirst for knowledge grew at an exceeding rate, driven as she was by the knowledge that magic existed in a world just outside of her reach.
In the time Hermione spent with her mother she learned of as many things magic as she could, although the route to the answers she sought was perilous. On good days, she learned of wands, spells, alleys and creatures. On the bad, her questions left her mother caught up in her memories, unable to discern what was real. On the worst days, Hermione comforted her mother as she panicked, screaming that she would never, could never escape him. Over the years, Hermione was able to piece together her father's story with carefully placed questions as she navigated her mother's rapidly deteriorating mind. She knew enough of the magical world to fear him, and to fear those who would find out her existence.
Her mother did not want her to attend Hogwarts. She believed it would result in her daughter's eventual discovery. She suffered from panic attacks and fits, so overwhelming paranoid and afraid to allow her daughter into the world of magic. But when the letter arrived at the Granger's home, Hermione was insistent. While she was properly terrified of discovery, she also felt the overwhelming need to escape her mother's demons before they consumed her as well. The distance was necessary for her own sanity. She detailed a plan and set it in motion.
So that was why Hermione sought out Harry Potter on the Hogwarts Express and befriended him whilst projecting a know-it-all persona as a Muggle-born witch. That was why Hermione ended up in Gryffindor, due to a ridiculous amount of begging on her part and some sympathy from the Sorting Hat as it acknowledged her bravery did indeed compare to her cunning and ambition. That was why she hid her disgust for the bumbling and insulting Ron Weasley and sought his friendship as well. That was why she endured the blood slurs. That was why she was the top of her class. That was why she played the part of the Gryffindor princess, because who would suspect a girl like that would have a father like that.
Hermione played the part perfectly up until the end of fourth year, when it all came crashing down around her. The Dark Lord had finally returned. She needed to warn her mother.