I thought it would be nice to have this up, I'm sure most people know by now, but if not, here you go. This is the method I used on mine, and I had it in about 20-30 minutes. First thing, your Eevee will need to know a Fairy type move. If you are catching an Eevee for the sole purpose of getting Sylveon, go after the level 19 ones found on Route 10 in X and Y. The level range for wild Eevee is 19-21. At 19 it will know Baby-Doll Eyes, but the level 20 and 21 Eevees will have this move replaced, with Swift I believe. Eevee does not learn another Fairy type move until level 29. Once you've got your Eevee knowing its Fairy move, take it into Amie. You've got to max its Amie Affection stat. Some sites have stated that you need to have all Amie stats maxed, but this is not true. Only Affection. Pet it until it responds with a music note, and then play Head Bounce. It's best to unlock Unlimited, which is done by beating Head Bounce on the hard difficulty. Stocking up on Puffs is the goal, specifically those with swirly icing, which raise affection the most. High combos in Unlimited will give you 6 or more each game of Head Bounce. After you have a decent number of puffs, feed them to Eevee. Again, the ones with swirled icing work best. Feed it until it will not eat any more, then pet it until it responds again with the music note. Then, repeat the game. After the game, feed and pet it, and repeat this until the Affection stat is maxed, which you can check through the 'Switch' option on Amie. This is the method I used, and I found it very efficient. If anyone has a more efficient method, or any tips, feel free to add them to the thread.
XD all I had to do was feed Eevee lots of treats and then level it up. I did this in about 5 minutes.