Well, this is a guide to getting Espeon and Umbreon. It actually will help with any pokémon you're going to evolve by happiness, but this is specifically about Espeon and Umbreon. Also this guide assumes that you're evolving the Eevee you got from Bill, not one that's hatched from an egg.
First off, it's best that you know that it doesn't matter at all when you train Eevee or give it haircuts. When it's time for it to evolve, it will evolve depending on the time then.
Now, when you're giving your Eevee haircuts, save in front of the desk and if you want to save time, put Eevee first in your lineup. Then talk to the haircut brother and let it give Eevee a haircut.
If it says "EEVEE looks a little happier", its happiness went up by 1 point. Press and hold Start, Select, A and B and the game will restart. Try again.
If it says "EEVEE looks happy", its happiness went up by 3 points. Press and hold Start, Select, A and B and the game will restart. Try again.
If it says "EEVEE looks delighted!", its happiness went up by 5 points if it was the older brother and 10 points if it was the younger brother. Save.
Yes, you heard me right. The younger brother gives more happiness. BUT, you may only have to restart a few times if it's the older brother, while it may take ages to get the younger brother to give you 10 points. Still, eventually, the younger brother will give your Eevee ten happiness points.
Now, to train Eevee, I've discovered a most interesting method. Go to Blackthorn city and down to the grass and save. Give Eevee a Smoke ball to hold. Search until you find a Geodude or Graveler, otherwise it's safest to run. When you find a Geodude or Graveler, use Sand-attack. Keep using it until they use Selfdestruct. If you're lucky, they miss. Otherwise, just turn off and back on. If they did miss, they will give Eevee loads of experience, then just save and find another one. It may take some time, but the experience and safety of this method is well worth it. Just don't forget to save after each Geodude or Graveler you manage to defeat.
Also, remember to feed Eevee vitamins!