The main goal here is to have Kabal Lackey and Mana Bind in your opening 4 cards. This will let you bring him down from 60 to 31 armour/health, giving you a monumentally larger chance of defeating him. This is better than the Lackey and Counterspell combo as although that lets you keep your full health, you'll still hit a brick wall at the Frostmourne phase. You need to kill him before his seventh turn, and instantly removing 29 of his health is a pretty good technique.
Failing that, you need Vicious Fledgling on Turn 3. Attack the face (as long as you can keep yourself alive) and adapt him for Windfury, more attack and protect from spells and hero powers in that order.
In either case the rest of the deck consists of taunt minions and freezing spells to stall while you attack his face, as well as damage spells (and Polymorph) to remove problematic minions as well as to finish him off.
Replacements: Babbling Book can be swapped for Shimmering Tempest, and Pompous Thespian for something along the lines of Bilefin Tidehunter or Sunfury Protector. There's no getting around two Kabal Lackeys, Mana Binds and Vicious Fledglings.