It's very random.I once got a pic from a villager about two weeks after they moved to my town but other villagers have taken as long as a year and a half.
I think some personalities are easier to befriend like normals and peppies, while snooties and crankies are a little harder. I guess that would factor into it as well.
it varies, I got one from one of my villagers after only being them just moving in, and a week later, after talking to them like a few times, I got their picture, and then another villager it took me three weeks of non-stop talking