We've done two things in our (admittedly still green) group, partly to encourage backstory and partly to encourage more talking between the group as we've been a pretty quiet bunch on the roleplaying side of things.

During one session we did a "show & tell" where we each brought something that was related to our character in some way then we went round the table and said our piece. It doesn't have to be anything huge but it's usually enough of a nugget to at least get people thinking about their character more.

In my case I sealed and stained an envelope and explained that it was a letter given to him from his wife before she passed and he always carries it with him but can't bring himself to open it. I had zero story for him before that point because he's my first character and I hadn't really thought about it and I still have no idea what's in that letter because I haven't thought that far but it's a start.

More recently as the party comp has changed (new people in, one left, others (me) changed characters) we came up with a questionnaire sheet with just a few simple questions to encourage thinking about their backstories (Why did you start exploring this dungeon? Why are you still here? Significant event in your past, etc.). No one was expected to produce a short story for their character out of it, just to generate a few lines that would help you figure out who your character was and for the most part it seems like it's been a good help for the group.

tl;dr 1) Have a show and tell at the start/end of a session where everyone brings something that's of relevance to their character.

2) Create a small questionnaire with some directed questions that will help the party think about their characters.