This achievement is for beating the DLC Revenge campaign, which you need to beat the regular DLC campaign to unlock. The easiest way to beat this campaign, and anything in this game, is to simply use the Exalted Darkness deck. If you still have a key, or don't mind spending 80MSP, you should unlock all cards for it to power it up a bit.
Exalted Darkness is, in my opinion, the most powerful deck in the game due to one card: Silent Arbiter. This card's ability is that it locks down the battlefield and makes it so that only one creature may attack and only one may block, which plays strongly into the deck's Exalted focus (which means that if an Exalted creature attacks alone, it gains +1/+1 for each other creature with Exalted that's under your control).
With this deck, matches may start out a little slowly and the opponent may take you down to 7-8 life fairly quickly, but once you get that Silent Arbiter in play, the match is basically over as you will control all battles from then on. The goal is to get one Silent Arbiter out as soon as possible, with a few Exalted creatures surrounding him. After that, just keep using your strongest Exalted creature - Aven Squire is a great card to attack with as it's a flying Exalted creature. Don't waste your early creatures trying to prevent damage as you'll want their Exalted boost for later. Just eat the damage and carry on until you have the Silent Arbiter/Exalted strategy in play.
Another useful strategy with this deck is combining Tormented Soul (which is unblockable) with Ring of Xathrid (which gives one black creature +1/+1 each turn).
If you're struggling with any opponent, you can always just restart the match until you get a Silent Arbiter in your first hand. That'll give you a huge advantage early on. You won't need to worry about finding Exalted creatures, as nearly all creatures in the deck have the ability.
UPDATE: This doesn't really work against the last three opponents in the Revenge campaign, especially Rakdos' insane deck. You're better off using a faster, more aggressive deck for those fights; I went with Dream Puppets and really only struggled against Rakdos. His deck is very fast, he gets ridiculously "lucky" with the draw and seems to have a counter for almost every strategy, so just keep at it until you get as lucky as him. Dream Puppets will work very well against Izzet, but might be dangerous against Golgari (due to Lord of Extinction). Try to keep Mind Control, Clone or unsummons on hand to stave off Lord of Extinction.