Evolving Eevee into Sylveon means maxing out your bond on Pokemon Amie.
This is quite simple in itself. All you need to do is shove Eevee in the Pokemon Amie, keep patting it and get it to the 'Switch Pokemon' screen when you aren't in Pokemon Amie.
(This screen:)
If the 'affection' is 5 hearts, then level up your Eevee a level and watch it evolve into Sylveon!
If it isn't, just keep patting it to it's at five hearts. xD
Hope I helped. :)
Source: Experience and this
