Use vitamins (can be bought in Laverre City)
Get massages once a day in Cyllage City (house directly to the left of the Pokemon Center)
Battle with it without letting it faint
Have your Eevee hold a Soothe Bell
Catch an Eevee in a Friend Ball (this will automatically set happiness to 200)
Get Colorful Shakes at the Juice Shoppe in Lumiose City (mix three berries of different colors)
After doing some of these, talk with the happiness checker girl in Laverre City (she's in the Pokemon Fan Club building). If she says "It looks really happy! It must love you a lot." the friendship is at max and your Eevee can be evolved! Level up your Eevee in a battle or with a single rare candy anytime after 8PM and before 4AM and you'll get your Umberon. Good luck! :)