I've beat the Dog with a spearazon, a claw assassin, and an axe barbarian. I'm chicken to try spell characters, 'cause those things DIE lots in early levels.
My cold sorceress took a while to get going, but once she got Blizzard she just ripped through everything in Normal and Nightmare. I've done twenty or thirty Nightmare Baal runs and she's never come close to dying. One cast of Blizzard will do about 7,000 points of damage over four seconds, which even the Nightmare Bosses can't deal with.
But...
Hell is a different story, since about half of the monsters are cold immunes. She has a level ten Fire Wall as a backup, but it kills too slowly. I struggled up to level 2 of the Claw Viper Temple, and every single monster there was cold immune - I couldn't even step down the stairs without getting surrounded and instantly killed by those slimy blue bastards. So now I'm stuck - I need to level her up some more (she's at 77) to work out a way deal with those Claw Vipers.
The only time my cold sorceress was in danger early on was against that Amazon chick (Blood Raven?), since she moved really fast relative to my shot speed. Aside from that it's just a matter of not getting surrounded, so I much prefer cold to fire or lightning for the freezing effects.
Oh, and Duriel kicked my ass, but that's mainly because it's the one spot where my computer slowed to a crawl.