Well, in this fanfiction, Naruto being exiled might be due to some false accusations. His kids could be born during his exile. He'd have to protect them from the threats that might come his way because of his exile. Maybe he trains them in secret arts that he discovers during his time away from his homeland, and they end up becoming a small but powerful family unit.
In a 'Naruto exiled has kids' fanfiction, perhaps Naruto is exiled because of a political conspiracy. His kids would then be born in a foreign land. Naruto could use his exile as an opportunity to give his kids a more peaceful upbringing, away from the wars and power struggles of the shinobi world. He might also pass on his knowledge of the nine - tailed fox's power in a new way to his children, and they could develop their own unique fighting styles based on it.