To encounter Darkrai, you'll need to have the Member Card, which will be distributed via Mystery Gift between April 1 and April 30. Once you've obtained the Member Card, travel to Newmoon Island, where you'll get a chance to capture Darkrai.
As with Arceus, you'll first need to enter the Hall of Fame and obtain the National Pokedex in Brilliant Diamond/Shining Pearl before you can encounter Darkrai. You'll also need to have traveled to Fullmoon Island and encountered Cresselia.
Pokemon Brilliant Diamond and Shining Pearl's 1.3.0 update went live at 6 p.m. PT on Mar. 15. In addition to Arceus and Darkrai, the update introduced the Global Wonder Station, a facility in Jubilife City where you can engage in Wonder Trades with players around the world.