I ask my own questions. Only I know what my story is about and what character traits are relevant to that story. When I try to ask generic questions that I have found on the web (e.g. "What is the favourite food of your character?"), the answers are usually meaningless to my writing.
I have looked at character software, but I found that their preconceived structures of character relations (e.g. protagonist-antagonist or hero-sidekick) or their pseudoscientific concepts of personality (e.g. the Enneagram) have more stifled my creativity and made my stories generic and unoriginal. I sometimes do use some lists of character questions I find on the web when I need inspiration, but once my imagination has been sparked, I quickly abandon them and let my mind roam free.
If you want to be original and creative, then using another person's concept of character will hinder you.
I take writing to be a dynamic process. I don't have to answer all questions about all my characters before I begin to write. I can always stop the writing and answer further questions, as they are raised by the advancing plot. I have found that I rarely had to rewrite anything I had written before because of these new Q&A.
To keep my characters organized I use a spreadsheet app (Excel), because it allows me to ask the same question (row) of all characters (columns) and see their different answers side by side and compare them. I can even easily highlight certain columns that currently interest me the most, or rearrange rows or include images. A spreadsheet app works better for me than a text editor (Word), where relations between pieces of content are difficult to visualize.
One method I employ to understand my characters better is to act them out.
I got this idea from drawing. When you try to draw a figure in a certain pose, it is sometimes difficult to translate what you see into lines of the page. To better understand the pose and how it exists in space, I often simply get myself into the same position (or, if I'm in a public space, imagine myself in that position, which works almost as well, because your mind remembers your body holdin certain poses). I can't explain how this works, but once I know my own body in the pose I see, I find it much more easy to render that pose on the page, often even without looking much at the model anymore.
In writing, I do the same. I actually often get up from the table and do what my character would do and say what they would say. Again, if I'm in a public place, I try to imagine acting it out as intently as possible. Interestingly enough this helps me better imagine what another person would do.
The book Getting into Character Brandilyn Collins explains this method. She adapts Stanislavski's technique of method acting to writing. I haven't read the book, so I don't know how good it is, but the description seems promising, if you need more detail and (probably) additional ideas.
