Instructions
Raise your Eevee's happiness to its maximum level. You can do this by having Daisy groom your Eevee in your hometown and by giving it haircuts in the Goldenrod Underground. Unfortunately, you will not be able to tell if Eevee is at maximum happiness or not.
Use your Eevee in battle once its happiness is at a high level or feed it a Rare Candy. This will cause the Eevee to gain a level. Make sure Eevee gains this level between 6:00 PM and 4:00 AM in-game time.
Select the "Yes" option when the Eevee begins to evolve. If the time is right and the Eevee is at maximum happiness, it will evolve into Umbreon. If the Eevee does not evolve, keep performing procedures that increase its happiness, then try to gain another level.