Instructions
Purchase several strong Poke-Balls, Awakening items/Blue Flutes and Reviving items in preparation for your encounter with Darkrai. Once you are taken to Newmoon Island, you will not be allowed to return to shop. Many of Darkrai's powers are more damaging when your Pokemon is asleep so it is important to have Awakening items to disspell this effect.
Travel to Canalave City and visit the house at the top right corner of the city, above the Pokemon Center. Presenting your Membership card will grant you access to this building. You will go to sleep after entering this building and then you will wake up at Newmoon Island.
Travel deeper into the island until you see Darkrai floating in the middle of a clearing. Save your game in case you lose to Darkrai, and then speak to Darkrai to initiate the battle.
Use your Fighting and Bug Pokemon against Darkrai, since it is weak against Fighting and Bug attacks. Avoid using Ghost and Dark Pokemon, since Darkrai is resistant to these types. Do not use Psychic Pokemon, which Darkrai is fully resistant against.
Capture Darkrai with a strong Poke-Ball, such as the Ultra Poke-Ball. This will be much easier if you can manage to put him to sleep before you attempt to capture him. If you fight Darkrai at night, the Dusk Poke-Ball may also prove to be extremely effective in capturing Darkrai.