I've tried befriending, ignoring, netting, and now the quickest way that I have found is with the "wisp" item, you can "summon" in a villager through amiibo cards and if you have ten villagers you can kick out any villager for the one on the card!
Earlier this week I acquired a little mouse named Bella, from my brother, who I didn't like to begin with but it didn't help when Papi, a villager I wanted, camped out in my town the same day Bella moved in! I wanted to get rid of her, so I tried the amiibo cards I had lying around from the release of Happy Home Designer, and she is now in boxes and is set to move out tomorrow!
This is a legit way to get villagers to leave, the only downside is you have to buy the cards and nfc reader (if you don't have the new 3ds), plus you're not guaranteed to get a good villagers in the card pack.
I feel so bad doing that! I saw that it was an option, but didn't want to even try! Then I got Stinky in a pack, and I love him! I invited him over to my house to get a gift from him, but I selected move in instead to see what the dialogue would have been. He told me how he couldn't move into a full town. Then, he said that we could always ask someone else to move out for him.... He said, "Like, maybe, Fuschia????" I was sitting there like, wow, how'd he know I can't stand her!? So I figured, Stinky's the one with the idea this time! I agreed to it, and got that deer out in favor of a villager I much prefer.
Stinky and the pink campground cow are the only ones I've moved in so far.
And remember... Bad times are just times that are bad
I actually used this to get all my dream villagers. I've got 291 Card out of 400 right now so I've got a lot of options and I love that I can ask them to move in even if the town is full because most of my original villagers were VERY annoying.