Gigei, I think, had a pretty okay childhood, considering she was sold and all. She did miss her mother a lot, her father... not very much, and then she had an almost normal childhood traveling through Tenkai. The way the troupe seemed to act was very family-like, so I imagine the children were taken care by everyone, and as the children grew up they started helping more. She loved the troupe completely.
Eagle was precocious like woah. Child-genius all the way, think Dougie Houser. I also think he was probably sickly when young thanks to Autozam's pollution, and he is an only child. In my headcanon his mother is a general in the army. He's close to his parents except that the three of them are insanely busy all the time. Eagle met Geo at the military school, maybe roomate? And Geo is a few years older than him.
Yuuko... well, before I always thought that she had inherited the store, too, but that's blown, but I do think that the granting-wishes is inherited. I think she and Clow knew each other through their dreams pretty much since forever, but they only met - physically, that is - once they were adults. Yuuko was obnoxious and loud and... well. Tsundere to the top when she was bitty.
And Kamui, I don't think I have anything in particular about his childhood. He probably HAS killed people before, but it's only recently that that started bothering him.