To have kits, you first have to choose which cat you want
For Dustfur, you would need too be a warrior, ask to battle, decline training, and a option for mating should show up; just say yes and you would be mates
For Duskfall, it's the same thing with Dustfur, except after declining, bring him to the shore and splash him with water multiple time, then the option should pop up
For Breezestar, you would have to be female, and I think just by talking to her you can get the option
For Nightwhisper, same as Breezestar
For Cotton (forgot her prefix), play guess my number around 3 times, get them correct, then ask to play guess my number again, decline, and the option should show up.
I haven't played this game in a while and I apologize for any mistakes