I already decided that the highest attribute would determine the class, but there is a tie for magic-user (intelligence) and thief (dexterity). I am choosing magic-user because the low strength would be less of a disadvantage for a quill-pushing magic-user than it would be for a law-flouting thief. So a spellcaster he is. His name will be... Phyrenzo, because it popped into my head.
Rolling All Dice, my results were d4: 4; d6: 3, d8: 8; d10: 2; d12: 8; d20: 19.
The d4: Family Reputation
With a roll of 3, the reputation of Phyrenzo's family is "Respected." We can assume that his surname, whatever it is, is a good one.
The d6: Personal Relationship
With a roll of 3, Phyrenzo in "Unappreciated." This may have something to do with why he would leave his respected family to go adventuring.
The d8: Number of Siblings
With a roll of 8, Phyrenzo has no siblings. This means we don't have to determine whether each sibling is a brother or a sister, nor do we need to determine his birth order, but it does mean he may have been a lonely child. He was probably a disappointment, hence the lack of appreciation.
The d10: Extended Family in Household
With a roll of 2, Phyrenzo grew up with a maternal grandfather in the home in addition to his parents. Maybe Grandfather knew something of the magical arts and provided attention that wasn't available in the form of siblings.
The d12: Family Occupation
With a roll of 8, Phyrenzo was born into a line of scholars of some sort. This fits well with his class. Maybe, however, his line of scholarship is at odds with the family tradition. Maybe his family is a line of religious scholars, possibly even clerics, whereas he and his grandfather felt a pull toward the dangerous study of secular magic.
The d20: Motivation for Adventuring
With a roll of 19, at least one of Phyrenzo's reasons for adventuring is the "spiritual quest." Maybe his respectable family of religious scholars turned him out with orders that he should not return until he has made his peace with their deity and given up dabbling in secular magic. Maybe they sent him on a pilgrimage in the hope that the journey will cause him to witness the miseries of the world and recognize the vanity of his chosen profession. Or maybe, alienated by his family's religious views, he is undertaking a spiritual quest of his own volition to find answers to his own questions, and these questions might be answerable by exploring the ruins of the past.
Starting with a blank character sheet and some randomly generated attributes, we now have a magic-user who grew up as an unappreciated only child in a family of respected religious scholars with only his grandfather to share his interest in magical studies. Now he is on a spiritual quest, willingly or unwillingly, that will cause him to face dangers nigh unimaginable, but may also end in rich reward (literally, metaphorically, or both).
Not every player character needs this much background information, and sometimes it's enjoyable to generate the background as one is playing, but for those who need the extra nudge to help them visualize their characters, I think it's a useful tool.