i don't really understand how one can develop a character's history during the RP? unless you mean character development, but isn't that an entirely different matter? it sounds kind of backwards to write up a character's history to suit their development in the RP, i think. i usually have at least a good idea of my characters' origins, family and life events before starting to write, if any. that's not to say i make really elaborate backstories either though... it's usually no more than a basic family setting, unless i'm going for something specific (like in Railways), so they usually still have plenty of room to grow.
I suppose it's not really developing a character's history so much as deciding it after the RP has already started. Of course, you have to decide some barebone things like where they're from and stuff like that but... For example by not specifying, or thinking about it explicitly beforehand, I can throw in that a character's parents are dead or alive when the subject comes up IC, depending on what feels better. Or they might have had some kind of traumatic encounter with this or that as a child if I feel like it. They might be secretly filthy rich-- or filthy poor, and the subject may just have never come up. I give them a personality and that's basically all I decide before I go off into the big wide world. I decide things as I the RP moves along, based on what kind of characters are around mine, and what kind of background would most support the kind of person I end up RPing the character as.
I don't always do it this way-- but I've done it once before with interesting results so I've been trying it more frequently nowadays.