How do you get Eevee to evolve into Umbreon?

First give Eevee many haircuts and then train it for an amount of time. When it gets happy enough, it will evolve into either an Espeon in the day-time or an Umbreon in the night time.

Also, make sure you don't put Eevee in your PC. And give it items like Carbos until it doesn't accept any more.

Be very nice to it. Give it items to hold, use it in battle often, and have it groomed, and during nighttime (in game) battle it. If you were nice enough, it should evolve after the battle is won.

I'll take your thanks and pocket it. You need to get your Eevee to be really happy with you, train it, walk with it a lot, give it haircuts (certain days in the Goldenrod underground you'll find groomers), give it items, and don't let it faint in battle. Eventually, it will be happy enough, that when you level it up, it will evolve into Espeon or Umbreon. Make sure it is nighttime when it levels up if you want to get Umbreon.

Don't teach it any moves. Just train your Eevee with the moves it already knows and will learn on it's own. I will be teaching it Shadow Ball but not until it's evolved into an Espeon. Also the more battles Eevee wins the better, so if you battle Pokémon that will give you less EXP points the better. It'll take more time but it'll evolve faster so you don't miss out on Psybeam at level 36 with Espeon.