NB: I like to say that writing is a lot like the after life in The Good Place. Your story unfolds in linear time, but the process of getting there looks a lot more like Jeremy Bearimy--a cursive, recursive, looping loop of guessing, asking, discovering, changing your mind, writing, and revising. That's a long-winded way of saying that the order of these questions is a suggestion. You can start anywhere, and you will almost certainly answer them many times over on your way to discovering who your characters are.
What?
Specifically: What do they want? What do they need? What are they willing to do to get what they want or need?
These are the core, fundamental questions that will define how your character acts in the story. There are plenty of other "what" questions you could ask, but you won't know which ones are worth knowing until you've answered these, and taken a look at the next three questions as well.
Why?
If there's a question on this list you should never stop asking about your characters, it's this one. Once you know what they want, you need to ask why. And then you probably need to ask why again. Here's an example to illustrate what I mean.
Let's say we have a character who wants nothing more than to solve her sister's unsolved murder. We ask why, and the first, obvious answer is "because it's her sister." But that's not enough. Some people have good relationships with their siblings, some people have bad relationships, and for many it's complicated. So why specifically does our character want to solve her sister's murder? Because she was distant from her sister and feels guilty that she didn't make more time for her now that she's gone. Because solving her sister's murder is a distraction from other, less exciting responsibilities that have fallen on her plate in the wake of her sister's death. Because those responsibilities include taking care of her sister's orphaned daughter, and that responsibility feels somehow even more overwhelming than solving a murder. Because she this is her only way to feel in control of a situation that is cruel and unfair and by its very nature beyond her control--and she needs to feel in control.
I hope you're starting to see why "Why" is such an important question. Often when you answer one "why" you are confronted with another. Why does she find taking care of her niece so daunting? Why does she need to feel in control? Is it simply a side effect of her grief? Or is there a deeper need there that is emerging in the wake of this tragedy? Their father was very controlling and their mother was absent--perhaps that's why she feels the need to be in control. Or perhaps it's not about control, but about freedom. She worries that she'll become trapped in the same life as her sister if she takes on her sister's responsibilities, and so she plays detective in order to convince herself she's still free.
How?
While "what" and "why" are the two big ones, the ones you can ask and ask and ask and always learn something new from, I find that "How?" is a terrific question for understanding your character's personality and the way they interact with other characters or respond to specific events.
As with the question Why? I try not to answer these questions too generally. The goal here is not just to ask about high-stress situations, but specific high-stress situations that will occur or are likely to occur in your story. How does our woman who wants to solve her sister's murder react when someone puts a gun to her head? How does she react if someone puts a gun in her hand? How does she seek justice for her sister? Let's say she despises guns. So even if someone gave her a gun, it would be an object of last resort--she may even throw it away. Now, you notice, we have another "why" to ask, and that's a good thing. It's another inroad into who our character is and what makes her tick.
Where?
The fourth essential question I want to know about my character is: where? Specifically: where do they start, and where are they going? This is as much about what the story is, as anything else. I need to know their journey, even if it's less progress than stasis, less revelation than evolution, and not at all a straight line.
And of course, knowing where my character is going gives me more opportunities to ask "why" and "how" again. How does she get there? Why does this event, or that conversation, or that character drive her in that direction? Only now I'm not just filling in backstory, I'm discovering the story itself: the present action that needs to unfold so that the character's journey feels organic and earned.