Barb. T1 for 10 levels and T2 the rest of the way.
I am doing Barb again on HC with T2 right from the get go. I've probably lost about 3 Barbs already. However, all these deaths have taught me better to optimize my builds for both DPS and survivability.
What I found works really well early on for Barb is getting 2 points in Lunging Strike as soon as possible. Then put 1 point into Upheaval (its insanely powerful AoE, but you have train as many mobs as possible for maximum kill efficiency and round them up and it will 1 shot them on T2 even at 1/5 rank) then max out your Lunging Strike to 5/5 asap, after that max out your Upheaval.
Once both of those are maxed, you can kind of start going a bit more freedom with builds, but those 2 really help. I've tried Bash, its horrible for HC, but does dps and the stun is useless when there are millions of mobs around. Lunging Strike let's you attack from a distance, so you can use it as early on mobility tool as well, even on destrucbile environment to get away from shit. And once it's maxed out, it even hits the champions/elite mobs hard as well.
I did both dungeons while leveling between level 7 and 10 - Mercy's Reach and Forbidden City on T2 that are located on the top left side of the map. I still had to play careful obviously, but the build was good enough to kill the bosses and survive their hard hitting abilities.
I am starting to love the Barbarian now. I tried every ability so far, and Lunging Strike seems to be the best for your auto attack early on. Flay can be good, but it takes years for the bleed to kill a single mob. Frenzy is not good early game and relies heavily on getting 2 good 1 handed weapons due to dual wielding. With Lunging Strike you only need to focus on a good 2 hander.
Excited too see how far I can take this Barbarian!