rather than describe the elements we feel should be delved into for this process, we'll do it through an example. That of a female elven warrior. No idea who she is, or what she's done. It's all done on the fly.

It's well and nice for her to be a mistress of the blade, skilled with spears, pikes and halberds, able to reliably hit the femoral arteries on orcs from a kilometer away... But that only covers skills as a warrior. Is she good at tracking? Can she cook? What's her view of young children? How does she treat dwarves and humans? How will she respond if that human companion tries to seduce her? Does she know how to read? if so, is there anything she really likes reading more than other subjects?

In the end, there's so much one can cover to make a character in-depth. If the elf is 342 years old, then there's no way you can cover her entire life in the character bio. Nor would you want to. Plot hooks in a bio are fellow players and GMs' best friends for building story.

One sort of plot hook to include is that encounter with the powerful human mage twelve years earlier. You don't actually need to cover anything beyond defining whether it was a good, bad or neutral event. The other players or the GM(s) can use that to build story from. Maybe one of the other PCs or an NPC is the mage's son or apprentice? Perhaps the mage proves to be the villain of a story arc?

Her skill at cooking might come from her father, who was made extravagant meals for the High King back when she was a naught but a child, learned over the span of seventy years. Or it could be because her mother was dead and her father was a lousy cook who could burn a salad in making it, and she had to take care of making all the edible food for her family.

She might hate dwarves, on account of how they're dirty, smelly and live underground. Or she could see them as master artisans, frequently trading with them for metallic ores, used by a smith she's known for a century and a half. Humans might be almost as disgusting, or perhaps the polished attitude and manners of that human knight might be all sorts of interesting when compared to smelly dwarves and elven men she'd known for centuries.

We would also ask why she fights the orcs. The setting might tell us why, but it could also be that it doesn't. Even if it does, we'd like to include the reason why she personally fights them. Perhaps one raped (and later killed) her mother? Perhaps her mother was transformed into one through dark magic, and she had to slay her to protect her infant brother?

Cover enough events like these, and we've not only got a decent bio, but we've got a blueprint for writing down personality, perhaps even most of her goals in life. Everything is, after all, tied together somehow. Even if one element is tied down just by a single thin thread.