As you already may know, double elimination is when it takes two losses to be dropped from the tourney.
Because of this, there is the existence of a winners and losers bracket. An example will best illustrate how this should work....
Say you have a 32 man tourney there will be 16 games. As you know there must be a winner and a loser. You know winner of game X goes against winner of game Y.
Your losers bracket will basically run the same way as your winners bracket. E.G. Loser of game X will play against loser of game Y.
As more people from winners are sent to losers as the tourney progress the brackets will work themselves out, depending on how you set them up. (e.g. Winner of game 15 in losers will play loser of game 17 from winners, and since an equivalent amount of losers are entering the losers bracket as there are people being eliminated from the tourney altogether, things should work out, like I said, all depends on how your brackets are set up, even though most bracket makers can help with that too).
Near the end of the tournament this is Losers Finals and Winners Finals. The winner of Losers Finals will play the winner of Winners Finals in Grand Finals.
When doing Grand Finals, keep in the mind that the person from Winners finals has to lose two sets in order for the winner from losers finals to win the tourney while the person from losers finals may only lose one set before the person from winners finals is declared the winner of the tourney.