Instructions
Head to Goldenrod, save your game and go inside the home to the left of the Game Corner. You will eventually meet up with Bill, who will give you an Eevee. Restart the game if the Eevee isn't a female. Buy or earn 6,666 Coins in the Goldenrod Game Corner. Exchange the coins for a male Eevee.
Keep the Eevee you want to evolve in your party. Give it multiple bottles of every vitamin until it refuses to consume anymore. Go to Goldenrod and pay for a massage. Level the Eevee up often and give it berries.
Avoid feeding your Eevee any bitter food, such as Energy Powder, Heal Powder, Energy Root and Revival Herb. Don't let your Eevee faint too much in battle. Each time a Pokemon faints, its happiness value decreases by one. Keep in mind that if you trade your Eevee, its happiness will return to a happiness value of 70.
Go to Goldenrod and talk to the girl in the home behind the MegaMart. Talk to the girl and she will inform you about how much your Eevee loves or hates you. Your Eevee has maximized its happiness if she says, "It looks really happy! It must like you a lot."
Feed your Eevee a rare candy between 6:00 PM and 4:00 AM (in terms of in-game time), and it will evolve into an Umbreon. If it doesn't evolve, you must have done something to decrease its happiness.