Hearthpwn

This deck isn't cheap to make. 
For a good budget alternative, go here: http://www.hearthpwn.com/decks/231129-general-drakkisath-heroic-easy-win-very-little-rng

Also I don't have all the cards. If I could, I would add 1x Onyxia, 1x Mekkineer Thermaplugg and 1x Piloted Sky Golem instead of 2x Stormwind champion and 1x Holy Fire.

That deck has maybe a 30% win rate. This deck, probably close to 50% if played correctly, and that is the key.

Most people fail this challenge because they play poorly. 

1. Mulligan. Mulligan for any deathrattle guy or guy that generates value. So: Dr. Boom, Cairne, Piloted Sky Golem, Trogzor, Hogger, Sneed's Old Shredder. This will test for his kill spell. If he uses it, you still get a guy.

Keep BGH and Mind Control. Those are the two best cards in the deck.

2. GENERAL STRATEGY. Where most people fail with this is they try to set up a Kel Thuzad lock in the first 2-3 turns and then hope to not get wiped out by Twisting Nether or Drakkisath's Command. That is a bad strategy. Hold off on the KT lock as long as you can. How you want to play this is you want to just 2 for 1 him every single turn.  
Once you have enough things on board to kill 2 of any minion he can drop, stop playing minions or play deathrattle minions, otherwise you are overextending into Twisting Nether. 

3. CARD BY CARD STRATEGY

 Lightbomb: Lightbomb is best used when he has 3 guys out and you can make yours survive. So this will generally be when you have force tank max out and he has 3x whatever. Also really good when you have sylvannas out and he played a force tank max. 

Kel Thuzad: The AI will never attack you if you have a taun and Kel Thuzad in play at the same time. If you drop Kel Thuzad behind a taunt, you have two options: Go face, or trade each turn and try to deck him. Just count the number of removals he's used ( he's got 2x Drakkisath's COmmand and 2x Twisting Nether ) and decide what you want to do. If you set this lock up early, better to trade for his minions and drop deathrattles until he uses Twisting Nether. That way you will bring him to fatigue and guarantee a win. Also never get Kel Thuzad down to 4 or less health as Drakkisath will then Flamestrike your board, killing Kel Thuzad and starting his attacks again.

Big Game Hunter: I try to keep this exclusively for Force Tank Max

The Black Knight: This can almost only hit Lord of the Arena. Black Knight also survives Lightbomb.

Mind Control: Your most important spell. Keep this for the best targets like Force Tank Max.

Ragnaros: Drakkisath will generally not try to trade for rag, so Ragnaros can sit there and generate a lot of value, killing a guy each turn, which basically evens out the match as long as Rag is in play and you're lucky. Turn 2 ragnaros is a really strong move.

Sylvannas: If you suspect he's out of removal, drop sylvannas on an empty board. Hopefully you can steal a force-tank max while also killing either a core hound or a lord of the arena.

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Well that's about it. If you play correctly and don't just rely on RNG to win, this boss is very beatable and much less frustrating. Yes he will still crush you with Kidnapper and there's nothing you can really do about it, but once you learn how to play and use the right deck you won't have to spend 5 hours restarting this over and over hoping to draw the perfect opening hand of 3 cards that also doesn't get countered by any one of 6 cards he's likely holding.