The easiest way is with amiibo, but it sounds like you may not want to use them.
I've heard that the villager who moved in most recently can't ask to move out, or if they can, then the chances are much lower than the other villagers. So if the one you want to move moved in most recently, then that might be why you're having trouble.
For me, the best way to get a villager to move out is becoming friends with them:
Villagers who have been there for awhile are most likily to ask to leave because its in their coding to want to leave after being on an island for very long time. They've been with you, seen what the island has to offer, now like any tourist, they wanna move on.
Sherb asks to leave even though he keeps giving me his picture, but its just his coding to want to leave.
----------------------------------------------------
If you have a villager you have not spoken much too, or would like to leave, then crowding them in friendship is best. Bombard them with apples or any gift talk to them lots, but not to annoy. You do not want to annoy them
Friendship will max quicker with visits. Trust me, my side Rep got Sasha's photo sooner because of the constant visits.
Once their friendship has maxed, they have been there for awhile, made friends, and are more likily to ask to leave. But he's my boi, so he's not allowed to leave lol.
Its nothing you've done, its just how the game goes. So in my exprience, friendship spamming is the way to go.
