Instructions
Download the "Membership Pass" from the Nintendo WFC. Nintendo periodically releases various items on their WFC network for people to download. Check the official Nintendo Pokémon website to see what is currently available.
Travel to Canalave City and use the "Membership Pass" to enter the locked motel. During the night, you will be visited by Darkrai in your sleep. When you awake, you will find his location on your map.
Travel to the southern part of the city and talk to the sailor. He will take you to Newmoon island. On the island, you will see Darkrai. Talk to it to initiate the battle. As Darkrai is a Dark-type Pokémon, you should use Fighting-type Pokémon to weaken it. Once it's health bar is colored red, you should throw any Pokéballs you have at it in order to capture it.
Tweaking
Make sure you have the bicycle, a Pokemon with "Fly" or "Teleport" and that you have used the "Explorer's Kit" at least one time.
Fly to Jubilife City. Mount your bike. Set it to its "slow" speed setting. From your starting point right in front of the door to the Pokemon Center, take five steps east and six steps south (you can use the "Step Counter" app on the Poketch to double check yourself). Make a mental note of the square you are in; this is square "one." Take one step west (you should be just on the edge of the curb); this is square "two." Take one step south; this is square "three." Remember these squares. Go back to square "one" and head left until you hit the building. From here, the goal is to ride, as quickly as possible, over the squares in this order: one, two, three, two, and one. This must be done very quickly, without stopping, and without hitting any other squares. If done correctly, you will be able to ride north to the Jubilife TV station and, when heading left along the curb, you should see that the entire west side of the city is missing and has been replaced with a grassy area.
Stand on the curb of the city side right before stepping off into the grass and reset the Step Counter on your Poketch. Take 20 steps to the west, 10 steps to the north, 5 steps to the east and 3 steps to the south. Open the Pokedex, exit it and go one step south. From this point, make 1 step south, 5 steps to the east and 430 steps to the north. Save the game. Reset your Nintendo DS.
Start the game again and travel 1,337 steps to the north, 65,408 steps to the west and 63,513 steps to the north.
Set foot on the island and open any menu, then press "B" to back out. Save the game and go into the forest to locate Darkrai.