Obviously, the details and style of any character biographies will depend on the characters and the settings. However, Indieproofing will provide as much or as little background as you require - just give us the details and see what we come up with.
Minimal, abstract characters can make a simple puzzle or adventure game more interesting. They can also help to add more depth to minor characters in a bigger game. The following description is based on a single image; more detail would be possible with a game mechanic behind the character.
Image courtesy of http://keeponstalking.blogspot.co.uk/2012/02/blah-blah-blah-blob.html
This is Blob. He's a shy, introverted sort of amorphous entity, preferring to stay home and tend his geraniums rather than go out and party. But Blob has a destiny - a big, blobby destiny that is already calling him to new heights of blobbiness. It's time for Blob to break out of his quiet life and make a name for himself. Or maybe tomorrow would be better...
For more complex games, you need more complex characters. Entire back stories may be needed for the main characters in an RPG - and, if you have a particularly complex world, these may interact. Similarly detailed descriptions could also be made for settings, historical events... give us as many details as you can, and we'll do the rest. The following description is, again, based on a fairly random image and much more could be done with additional details.
Image courtesy of http://wallpaperscraft.com/download/pirate_fire_sword_man_3602
Born to lowly parents on a cold winter's night, Hastor's life was never going to be easy. Fortune blessed him with a nimble body and an equally nimble mind, and Hastor quickly became skilled at using both. But fortune takes as well as gives, and Hastor was barely past ten summers when bandits from the neighbouring kingdoms put his parents to the sword and set his home ablaze. Hidden from sight, all he could do was watch as his life fell apart.
With no home and no family to fall back on, Hastor took to stealing in order to survive. His quick fingers could often find their way into an unguarded pocket or liberate a forgotten trinket from the local store while the owner wasn't looking, though as he matured he found his quick wits worked even better. He learned to fight on the streets, for there were times when a sharp knife was a better defence than a sharp mind. And if the city guards got too close - well, there were always other towns.
The teenage Hastor took things too far, however, when trying to remove some pretty baubles from one magical store. The bumbling old wizard who ran the shop was a lot less bumbling than he appeared. Yet Hastor was in luck - rather than turn him into whatever creature first popped into his head, the old man decided to take him in and try to reform him. While Hastor's thieving never quite stopped entirely, he did his best to limit it when given the promise of three square meals a day and a bed to sleep in.
Working with his new mentor, Hastor developed an interest in magic, discovering an aptitude for simple spells, particularly those that could be useful for opening doors or spiriting away valuables - honesty still didn't come easily to him. However, a chance meeting in the marketplace changed his focus - even after all these years, the scarred face of the bandit chieftain that had killed his parents was instantly recognisable. Hastor took to magical studies with a new aim - vengeance.