Once you know these key internal things about the character, you can start adding other character traits. Think about how the character fits into her world -- does she fit in, or is she a misfit? Does she like her current situation? Who are the people in her life, and how does she relate to them? Where does she live, work, play, and what does she think about these things? I think the character's attitudes toward the things in her life are as important as the things themselves. There are few things in our lives that we don't have some attitude about, and a character without opinions and attitudes about the details of her life will come across as weirdly blank. For instance, even if you don't think about your neighbors very often, you probably have attitudes or opinions about them -- they're noisy, intrusive, weird and reclusive, messy, leave their trash bins out, don't clean up after their pets, are way too obsessive about their lawns, come and go at odd hours, are like the family you never had, etc. So, it's not just important to know your character's profession, but you need to know what she thinks about that profession.
I once went to a writing seminar where the instructor suggested that each character have four key traits -- like stubbornness, altruism, enthusiasm, etc. -- and that one of them should be something that doesn't seem to fit. Too few traits and the character seems undeveloped, but more than that and the character will be unfocused because you won't have time to really develop all those traits. I've never actually made that work, at least not in a first draft. I find that I'm more likely to discover traits like these along the way because they emerge as I write the characters, and then I can work to develop and emphasize those traits in later drafts. I do think that having one thing that doesn't quite seem to fit, that's unexpected, plays a huge role in creating a character that people find fascinating. A person who isn't entirely what he seems to be or what you'd expect him to be is automatically intriguing. This oddball trait shouldn't come out of the blue, though. It needs to make sense for the character, and you may have to do some digging into the character's backstory or inner life to either find the oddball trait or figure out how the oddball trait fits. In my series, the character who most captures people's imagination is Owen, the super-powerful, very handsome wizard who's also ridiculously shy and can barely have a conversation without turning bright red. There is a story reason behind the shyness that I know and that's only hinted at in the books, and I think that oddball trait is one big reason why readers are so intrigued by him.
How much backstory you need to develop depends on the character and the story you're telling. Some characters and stories may require you to at least think about all the major points in the character's life, even if you don't mention those anywhere in the book. Some characters only need the slightest amount of backstory because their past doesn't matter that much. I would say to develop what you feel you need to understand the character, but remember that you don't have to put any or all of it in the book. If you need to know what his elementary school years were like to grasp where he is today, then think about that, but don't feel you have to develop his whole life story if you have a good sense for who this person is and that information doesn't matter to the plot. I find that there's a point where a character clicks for me and I feel like I know this person well enough to write the book. I'll always discover stuff along the way, and that's what revisions are for.
Next I'll get into how to convey all this character development in a story.