Vaporeon: Surf, Ice Beam, Aqua Ring, Shadow Ball
Jolteon: Shadow Ball, Thunderbolt, Thunder Wave, Thunder
Espeon: Psychic, Shadow Ball, Grass Knot, Dazzling Gleam
Flareon: Flamethrower, Flare Blitz, Shadow Ball, Return.
Sylveon: Shadow Ball, Moonblast, Calm Mind, Psyshock
Glaceon: Shadow Ball, Hail, Blizzard, Mirror Coat
This is a pretty standard Vaporeon moveset: Surf and Ice Beam are musts on any special Water types. Aqua Ring will restore HP and Shadow Ball for Coverage.
Jolteon does not have a lot of coverage. You could technically use Hidden Power Ice, but I don't like trying to calculate Hidden Power type and power so Thunder and Thunder Wave made more sense for me. With Jolteon, you are getting a true specialist.
Espeon: Psychic and Shadow Ball are pretty standard fare for a Psychic type. Dazzling Gleam will take care of Dark types, and Grass Knot will take care of the Water/Ground and Water/Rock types (Quagsire, Whiscash, and Swampert are relatively heavy so it will be pretty useful).
Flareon: This doesn't make full-use of Flareon's 130 attack stat, but it's 95 special attack is quite good. The thing with Flare Blitz is that it damages yourself so I'm less inclined to use it. Which is why I have Flamethrower and not Overheat here. Shadow Ball for coverage, and Return for pure power (130 base attack). If you can breed Iron Tail onto your Eevee, however, replace Return with Iron Tail!!!
Sylveon: Sylveon can deal with pesky Poison types with Psyshock, and since Steel no longer resists Ghost, this should work okay against Steel. Since its special ability is Cute Charm, you do NOT need Attract.
Glaceon: Its special ability is Snow Cloak, and so Hail is a must. Shadow Ball again for coverage. And when there's Hail, you HAVE to have Blizzard. Mirror Coat will send back all special attacks at double the damage.
Take this moveset with a grain of salt, however. If you can breed Fake Tears onto your Eevees, then replace Vaporeon's Aqua Ring, Jolteon's Thunder, and Sylveon's Calm Mind with Fake Tears. Fake Tears followed by a special move would crush all opposition.
Maybe you could try using an Umbreon. I'd guess that people would try to use Double Team, Toxic, Confuse Ray, and Attract/Screech/Mean Look/Moonlight. If I wanted to, though, I'd still keep Toxic and Confuse Ray, but maybe I'd use Fake Tears and Dark Pulse or Screech and Faint Attack. You won't be doing much damage with Umbreon, but he'll take enough hits for you to get in 3 Fake Tears/Screech without Attract causing immobilization. Taunt and Payback/Assurance are also good fits.