As I'm sure you've all noticed by now, Isshin takes his camp job pretty seriously. He got hired to be a daddy to you all and he's damn well gonna. But why is he so enthusiastic about it? I'm going to try to explain this part of my characterization of Isshin in a way that makes sense.
In the third volume of Bleach we get to see Isshin being simultaneously calm and serious for the first time. (Er, mostly.) When he's telling Ichigo that he fell in love with a woman who would die to protect her son, he's smiling like that's the best sort of person to be [
1]. Even if it would be easier to hate Ichigo for giving Masaki a reason to die, he can't. Isshin loves his son too much and, as we later see during the story with Don Kanonji, he can't exactly say he wouldn't have done the same in her position [
2,
3]. There's no way of telling how he came to have this sort of dedication until muuuuch later, when all of a sudden we learn that Isshin used to be not just a shinigami, but a shinigami captain. Shinigami are (supposed to be) dedicated to protecting the living and the balance of souls between the living world and the dead. For a captain who believes this whole-heartedly, it's only reasonable that if you protect perfect strangers with your very existence, it's even easier to do the same for the people you love.
Given what we know of Isshin from canon, it would be safe to assume that he still wants to help people when they're alive or he wouldn't have decided to become a doctor. There are plenty of other ways he could've chosen to support himself and his eventual family (and hey, even keep tabs on Ryuuken, since it really seems to me like they knew each other for a while before Isshin decided to play human full-time) otherwise. Between the two, Isshin sees all the kids in camp who haven't had the benefit of coming from a family as devoted as the Kurosakis and can't help but want to show them what it's like from the inside. And, because he got the job of Official Multi-Purpose Father Stand-In and has been playing Kooky Daddy for years, it's pretty easy to just point at someone and say "you! Adopted!" even if he doesn't necessarily know the person's name.
And that's basically it. I'll see if I can't flesh it out later, but if you have anything you want clarified or just want to discuss, have at.