Instructions
Make your way to Hearthome City and find Bebe's house, near the Pokemon Center. Speak to her, and she will give you an Eevee.
Raise your new Eevee's happiness by providing it with berries and having it groomed. You can also earn Eevee's affection by placing it first in your Pokemon line-up.
Go to the Pokemon Fanclub, also in Hearthome City, and talk to the Pokemon happiness rater to find out your Eevee's happiness level. When the happiness rater responds with "It is super friendly to you! I'm a bit jealous!" you can now evolve your Eevee.
Increase Eevee's level during a nighttime battle. Be sure the game's time clock is between 8:00 PM and 4:00 AM. Once Eevee levels, it will evolve in Umbreon.