I just posted my method in another thread, I'll post it here too:
Definitely talk to the unwanted villager a lot, I have noticed that villagers are more likely to try to move out if I talk to them a lot. Also, if you haven't talked to a villager in a week, they will not ping you. So if you're trying to get a villager to ping you and they don't, talk to them to make sure you're on speaking terms.
If you're not opposed to time traveling, it's definitely the fastest way to get villagers out. I had an unwanted villager for some time and it took me awhile, but I finally got her out after a few (real life) days of TTing, with over 15 requests from my other villagers to leave. I was able to get into a groove that worked quickly and efficiently, with 0% chance of accidentally moving out a different villager. In fact, never in ACNL have I accidentally lost a villager. Here's what I did:
1. Make sure to have beautiful town enacted so flowers don't wilt/weeds don't grow
2. Check if a villager is currently trying to leave. Talk to 3 different villagers all the way through until they won't talk to you anymore; one of them will tell you if someone is trying to leave (note: I've heard that if the moving out villager isn't awake, no one will tell you they are leaving. Make sure you're talking to villagers when everyone in your town is awake. The last to wake up are Uchis at 11am. I usually do all my TTing at noon to be safe).
3. If none of them say someone is leaving, TT to the next day (I always TT using the 3DS settings, not by talking to Isabelle. If you get too comfortable and go to fast, it's possible to accidentally delete your town. Using the system clock, there is way less risk. Just remember to set it to the next month if you get there, as going from July 31 and setting the day to 1 will not automatically change the month to August.)
4. Repeat step 2. If no one is leaving yet, repeat step 3. Repeat steps 2 and 3 until someone is leaving.
5. Once someone is leaving, if it's not the one you're trying to move out, get them to ping you and stop them from leaving (note again that if you haven't talked to them in more than a week, they will not ping you).
6. Now that you've stopped someone from leaving, it's time for some time travel. Go forward 3 days, open the game, save and quit.
7. Now TT again, this time only 1 day into the future.
8. Now talk to the villagers again to see who is moving. If no one is moving, TT one day into the future and check again.
9. Repeat steps 5-8 until your (un)desired villager is leaving.
10. Once you find out the villager you're trying to ditch is leaving, save and then try to get them to ping you. If they don't ping you at first, talk to them and them save, quit, and start up the game again. They should ping you then and you can accept them leaving. Make note of what day they are leaving.
11. In the event that the villager decides to not leave after you talk to them, close the game without saving (power off or home menu and close). Open up the game again and try to get them to ping you again. They will say they're moving again and you can try to get it to go through. Make sure you don't save until they say they are leaving for sure!
12. Now that your undesired is leaving, make sure not to talk to them. I heard they can change their mind and decide to stay after all. To be safe, you can TT to the date they are leaving to make sure they're gone for good.
The reason for TTing 3 days and then 1 day is that there is zero risk of a villager fully moving out in that time (provided you just stopped someone from leaving). Once a villager decides to leave, they will move out in 5 days. So they'll have a day left if they decided to move out immediately after someone else (which they usually don't anyway, but it's good to be safe). I found the most success with this number of days. Using this method, after every 3 day-1 day cycle, someone would try to move, without fail.
