How do you like to start a campaign

[Distracted DM]

As @prabe noted, many games provide for this in their toolkit. Over the Wall does it during the life path character building, with a lot of "turn to the player sitting here and ask.." that you could easily incorporate into other games. Others provide background questions that may or may not be somewhat provocative. Eg: Daggerheart has teh following Connection questions for the rogue to ask others:

• What did I recently convince you to do that got us both in trouble?
• What have I discovered about your past that I hold secret from the others?
• Who do you know from my past, and how have they influenced your feelings about me?

His Majesty the Worm mechanizes "Bonds" between Guild members (the dungeon delving party is formalized as a chartered Guild in the framing of that game), which include things like "Big/Little" or "Best Friend."

Apocalypse World's Sword and Sorcery hack "Warriors of the World Ablaze" has questions on playbooks like:

• Have any of you fought shoulder to shoulder beside me?
• Do any of you truly depend on me for protection?
• I admire practical power. Are any of you, more interested in abstract, mystical, or otherworldly power instead?

Yeah, these are the sorts of things I was looking for. I've improvised a few at session 0's but was never fully satisfied with them.
I don't see a collection of Warriors of the World Ablaze, just playbooks released on the designers patreon. Dang.

 

[Distracted DM]

That said, I'm running D&D 5e/Tales of the Valiant. There are other games that have more structured approaches written into their text.

Yeah, definitely looking for some structure to make use of. Some folk benefit from instruction, some guardrails, they don't jive with open-ended stuff as well.