She really isn't hard to understand. The thing to remember is that her mystery, her seeming impossible nature, is merely a quirk of the Doctor's time-traveling lifestyle and our seeing her story from his perspective. He, and we, encounter the consequences of her decision to save him by stepping into his timeline before we meet her younger self, Modern Clara, who eventually made that choice.
It would be very interesting to find someone who hasn't seen her story, and show it to them from her perspective. Start with "The Bells of Saint John" but edit out all of the Doctor's stress about her being impossible and the woman twice dead. What is it like to get to know Clara without that distraction?
What you have is an educated young woman, finished with university, and either working as a teacher or on a break from work, who is staying with some friends who recently lost the mother of the family, and whose teenage children are going through the same loss she experienced as a teenager whose mother died. She's compassionate, but clearly an adult to their teenage nature. She's not trying to replace their mother. She does make a point of being an example of someone who has lost their mother, but has dealt with it, and an example of how you can remember your mother lovingly, and how the initial pain of early mourning will change.
Clara is not working as a nanny. She's helping friends, temporarily. They are already looking for a longer-term solution to their needs when we first meet her. She's relaxed, trusting them to not make demands on her indefinitely. The father is grateful, and looking for other help so that he isn't taking advantage of her generosity unreasonably.
Clara is motivated by kindness and friendship. Professionally, she's motivated to educate, nurture, and mentor. These things drive her interactions with the Doctor and her choices in their adventures.
Clara's strength lies in her people skills. She's not good with computers. She doesn't get along with the TARDIS. And when she does interact with the TARDIS, she does so by treating it as a person. Characterizing the Doctor's need for the TARDIS to like her as being like a boyfriend who needs his mother to like a girlfriend. But she isn't really comfortable with the idea of a person-like machine, also stepping back and characterizing the TARDIS as an "appliance." Less impressed, and more uncomfortable with the alien than other modern companions have been. The hacking skills she picks up when uploaded are put to use not merely hacking, but by understanding how the people working for her enemy will have used social media, and using that people-knowledge to find them.
Clara has an interest in travel. This is related to her fond memories of her mother, who shared the dream of travel with her through the "101 Places to See" book. But this is a fond memory and keepsake, something of a scrapbook of memories as well. Clara is not obsessed with this desire to travel. She'll postpone travels to care for friends in need. She'll take up the Doctor's offer to travel with him, eagerly. But she's not running away with him. When he frames his offer as "running away" she firmly declines, and suggests he comes back in the morning and ask again. And while the "101 Places to See" book represents her interest in travel, it doesn't define it. She's not looking to work her way through the book. She wants to see amazing things. And that's what she asks the Doctor for.
Clara is very much an adult. She's finished school, she has a job, she has a career and life she likes. She is at a stage in life that Amy and Rory were only beginning to reach when they left us. And she picks up their mature travel habits with the Doctor where they left off. In a way, this gives the Doctor a chance to continue to grow in his understanding of what a companion can be. Rory asked why the Doctor can't just phone ahead. Clara insists that he does phone ahead.
While we got a clear view of all of what a companion's life was with earlier modern companions, with Clara, her life is largely off-screen. What we see is her vacations and holidays.
Clara has a clearer sense of her own mortality than earlier modern companions. She's experienced death, up-close, through the death of her mother when she was a teenager, old enough to understand. Her first instinct, in danger, is to retreat to safety. She's not looking for adventure. But she will step up, and not walk away when someone is in need. As we see with the friends she's helping, with her reaching out to Merry as a scared child, with her responding to the disaster that was the GI disrupting the Doctor's timestream by sacrificing herself.
Clara isn't driven to rewrite history. Rather, when knowing history gives her a glimpse into a possible future, realizing that she must have stepped into the Doctor's timestream to be the women he remembers, it gives her the courage to carry through the act, writing history rather than rewriting it.
In a way, she reminds me of Liz Shaw, another woman who had a career, and who liked the Doctor but would not let the Doctor disrupt her life or hold her back. It is playing out differently because Clara's career doesn't put her in competition with the Doctor, and the Doctor isn't holding her back professionally. LIz was a trained scientist who wouldn't play second fiddle or function as merely an assistant, and Clara rejects the role of "assistant" as well. There is also a fair amount of Rory, particularly early Rory in her, with her refusal to be impressed, and her willingness to question the Doctor. But while Rory confronts the Doctor, Clara more simply says "no" to him when he's asking for more than she's willing to give, and she makes her "no" stick.
Clara is also interesting in that she's often made uncomfortable by the things she experiences. She initially sees the Doctor wanting to take her into a small box rather creepy. She's freaked out when the Doctor takes his snapshots of the life of Earth. The things that the Doctor shows his companions aren't always easy, and it is refreshing to have a companion who takes time to think about what she's experiencing, and who questions the new things she's experiencing rather than running ahead unthinkingly.
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