Start by getting five Lucky Eggs. Put a pokemon with Compound Eyes at the head of your party and let it be knocked out, followed by a high-level user of Thief in the second slot, and go to a Friend Safari in which Chansey can be found. OHKO every Chansey you encounter with Thief and make sure to notice whether or not you stole an item. You'll get more Lucky Punches than Lucky Eggs, but eventually you'll have enough for your whole party.
Catch or hatch the pokemon you want to train on a different cart, or have a friend do it. Once they're on your cart, you may optionally get them to rank 5 affection in Pokemon-Amie. You can stop short of rank 5 if you want, but you should still have at least one heart.
Have them each hold a Lucky Egg. The leader of your party should be a high-level pokemon with good coverage moves with high PP and no double weaknesses. It is good if it is holding an Amulet Coin or Luck Incense.
Enter the Battle Maison at 12:01 AM. Issue a Black Writ of Challenge. A new trainer will spawn every few minutes. If you're about to fight a Furisode Girl, a gym leader, a member of the Elite Four, or a Champion, have a friend send you an EXP Power level 3. (Said friend's 3DS should have recorded at least 4000 steps recently, so the O-powers recharge quickly.)
Make sure the Exp. Share is turned on.
Each pokemon will get half EXP from the Exp. Share. This will be boosted by 50% by the Lucky Egg, by another 50% due to having been traded, and a further 50% because of affection. This amount will then be doubled by the O-Power. The end result is that each of those five is now earning triple the amount of EXP.
For 24 hours, battle every trainer in the Battle Maison. Checking on the state of the place every half hour or so should be sufficient. Bring a good TV series to watch.
