Instructions
Go to the Pokemon Center in Ecruteak City and speak to Bill. Follow him to Goldenrod to get Eevee for your party.
Use Eevee frequently in battle to gain experience with this character so it can level up. If Eevee is poisoned or damaged, heal this Pokemon quickly.
Increase Eevee's happiness by visiting the barber located in the Goldenrod Underground Tunnel. Check in with a woman in Goldenrod to determine how much Eevee likes you.
Level up Eevee one more time if the woman in Goldenrod tells you that this character's like for you is maxed out. Eevee will evolve into Espeon if this takes place between 5 a.m. and 6 p.m. in game time, or Umbreon if the evolution occurs between 6 p.m. and 5 a.m..