It's very simple!
To buy/adopt a villager the first thing you need is less than 10 villagers. (You could have 10 and try to TT until someone moves out, but let's say you need 9 or less).
Then you need to have an empty house plot. You can either buy one from Nook or use one that's leftover from a villager that's moved out recently (like, in the past day).
Once you meet these two requirements you can go visit someone's island and talk to the villager that's in boxes (moving out). You'll know they're in boxes because that's when they're advertised around here and are put on sale. You head inside their house, talk to them and convince to move to your town.
The next day they'll move into your empty plot!
As for the campsite, no, you yourself can't get villagers there unless you use amiibo cards. The rest is all random.
As for making someone move out, it seems to be completely random. Hitting them with your net is as helpful as talking to them in order to make them move out. I'm afraid all you can do is wait.
I tried to be as clear and concise as possible, hopefully I was of some help!