There is no straight up way to "make" an animal leave your town, but you sure can convince them!
The way I understand it, the reason animal stay is based on how close they are to you freinship-wise. If you're a jerk to them, they'll be more likely to leave. Here are some things you can do to make them like you less:
- Hit them with your net repeatedly. They will get upset after 3 hits and not want to talk to you
- Refuse to give them medicine if they are sick.
- Open an item/letter they give you to deliver, especially when they specifically ask you not to
- Deny their requests
- Fail their requests
- Ignore them for long periods of time
- Open time capsules that animals requested you to bury
One thing that does NOT work, but is commonly suggested is reporting them to Isabelle. The options are "I don't like their language", "They have a rude letter", and "Her clothes are wrong". This is because of the customization aspect of Animal Crossing: New Leaf, where people can customize catchphrases, clothing, and write letters. Because of the possibility for harmful content via animals from other towns moving to yours (I've had this happen once before). Isabelle will then remove whatever offensive material they had and replace it with something generic.
I know this to be the case, because a chicken in my town was walking around in a shirt I designed. For fun, I told Isabelle her clothes were wrong. The next time I saw her, she was wearing her default outfit!
Hopefully this list helps :)