When you catch Eevee, its happiness is 70. It evolves into either Umbreon or Espeon by leveling up when its happiness is 220 or more. Here are common ways to increase a Pokemon's happiness:
Walk 256 steps: increase by 1
Level up: increase by 5 if happiness is 0-99, 3 if 100-199, 2 if 200-255
Using Pomeg, Kelpsy, Qualot, Hondew, Grepa, or Tamato berry: increase by 10 if 0-99, increase by 5 if 100-199, 2 if 200-255
A better list can be found here.
Remember that fainting decreases the happiness by 1, if Eevee was caught in a Luxury Ball or a Friend ball, the happiness gained from leveling up increase to 6, 4, and 3, respectively, if the Eevee is holding a Soothe Bell, any increase is multiplied by 1.5 and rounded down, and the happiness resets when the Eevee is traded.