This page details the way I raise my Umbreons in the games... unfortunately there are no pictures, but I did my best to be thorough on this page. These methods are particularly for Silver, which is my most favorite Pokemon game version (it works with Gold and Crystal too, I would hope to imagine). That said, let the games begin!
I like to start fresh. As in really fresh. I breed the Eevee that Bill is so awesome for giving us with a Ditto; you can, of course, breed it with anything in the Ground egg group (as long as the Eevee is female - if it's male, breed with Ditto), but considering that Eevee doesn't inherit any moves that are too worthwhile upon hatching, it doesn't really matter that much. What does matter is how you treat this level 5 angel when it hatches from the egg 8,960 steps after you get it.
You must treat the Eevee like the awesome god(dess) that it is. Give it haircuts in Goldenrod's Underground Pass (don't bother with the younger brother, the elder is better). Feed it Carbos, Calcium and Protein to up its speed, special attack and attack (Umbreon's special attack and attack levels are low compared to its defense). Give it something to hold - DON'T give it anything bitter to hold, no bitter berries or anything - this will make the Eevee HATE you and it will take even longer for you to evolve it - which may or may not mean missing out on crucial moves. (Also, don't be an idiot and give it an Everstone, because an Everstone STOPS Eevee from evolving. Give it something like EXP Share or a Berry, and if it eats the Berry during battle, give it another one!) Don't buy anything from Granny's store in the Underground Pass - she isn't kidding when she says your Pokemon won't like it. And DON'T let Eevee faint often. Or if it DOES and you couldn't help it, revive it immediately with either a Revive or Max Revive- and if you can't afford them, fly to the nearest PokeCenter and make sure you buy super potions and status healers (Paralyz Heal, Burn Heal, Ice Heal, etc) at the PokeMart so it won't be so prone to happen again.
Generally, if you're using Bill's Eevee rather than starting over, Eevee will evolve in its 30's or 40's, which may or may not be beneficial since it does learn Bite, but misses out on Pursuit and Confuse Ray. Bite is stronger and your Eevee can learn it at level 37, but Pursuit is a powerful move for a Pokemon that's just starting out: Umbreon learns it at just level 16. If you start over with an egg and give Eevee an overload of love, you should have your Umbreon by then. Crucial: Try to save before each level Eevee obtains. If it tries to evolve during the day it will become an Espeon - but you don't WANT an Espeon, you want an Umbreon - and every level it takes to evolve it is important. If it tries to evolve during the day, turn off the game and wait until night to resume training your Eevee. This way you don't get an Espeon and you don't waste a level on it.
When your Eevee does evolve, click here to check out the moves that Umbreon will be able to learn as you train it. Pay special attention to Confuse Ray (level 30), Faint Attack (Level 36) and Moonlight (Level 52). Those are (in my opinion) the best moves that it learns on its own. Confuse Ray can help you take down a tough opponent by confusing it into hurting itself. Faint Attack is a weak Dark move compared to Crunch, but it's the best that Umbreon learns on its own aside from Bite if it evolved late. And Moonlight helps Umbreon restore its HP and is more effective during the night, as you probably could have guessed, but the stronger your Umbreon gets, the stronger the attacks are. Note how I don't list a fourth move - I usually go with Quick Attack until I obtain TM29, Psychic, at Mr. Psychic's house in Saffron City (you can get more in Celadon city's game corner for 3500 coins apiece). I love Psychic and I love teaching it to Umbreon because Umbreon is weak against Fighting types; if you use Psychic against them you'll blow them away. The only other type that is super effective against Umbreon's pure dark type are Bug types, which can also be dealt with using Psychic.
And that's about it, really, that I can think of, other than give your Umbreon Blackglasses to hold (you can retrieve them in the Dark Cave) to boost the strength of its Dark type attacks. I hope that this little guide has been helpful to you, and good luck!