I don't think it's possible to win on turn 2. I can't think of anything that could happen.
Edit: I forgot about Millhouse and on turn two, you can have six cards if you go second.
It's possible with rogue, going second and drawing:
Young Dragonhawk, Cold blood, Cold blood, Abusive Sergeant, Coin
Turn 1: (Draw shadowstep)
Young Dragonhawk
Turn 2: (Draw shadowstep)
Coin
Cold blood (on Dragonhawk)
Cold blood (on Dragonhawk)
Abusive Sergeant (+2 on Dragonhawk)
Shadowstep (on Abusive Sergeant)
Abusive Sergeant (+2 on Dragonhawk)
Shadowstep (on Abusive Sergeant)
Abusive Sergeant (+2 on Dragonhawk)
Attack twice with Young Dragonhawk which is now a 15/1 minion. 30 damage
BONUS (the below will never ever happen ever)
It's also possible with Priest when playing against a Rogue where the Rogue has two Preparations in hand, and a Ysera in his deck, and the priest has:
Mind Games, Inner Fire, Mind Vision, Divine Spirit, Coin
Turn 1: (draw Mind Vision)
Coin
Mind Vision (get Preparation)
Preparation
Mind Games (get Ysera)
End of turn: Ysera draws nightmare
Turn 2:
Nightmare on Ysera
Mind Vision (get Preparation)
Preparation
Divine Spirit (on Ysera)
Inner Fire (on Ysera)
Hit for 34 damage
I can't take credit for the Priest combo, it came up as a response to this reddit thread I posted a few months back asking the same question (but rather if any class except rogue could do it) here:
http://www.reddit.com/r/hearthstone/comments/22d48w
Other than those two, I'm fairly confident that the only other possible turn 1/2 kills are with Millhouse, but that's kindoff cheating