I've had success with concentrate and 2 handed weapon. But keep 1-h + shield on switch if one should need resistances, and accept the lower damage.
Probably anything can work. It is the play style that matters. Double swing is fast and high damage, but you'll need high defense in order to tank damage. And that can be difficult. Waking or standing still when there are many monsters about can save your life.
One fun build is double throw. Barb throws exceptionally fast, and it deals high damage too. Makes boss fights easier. You can even use a bow against mephisto and moat trick. It's super slow but safe. I think throwing spears reach him from that distance, can't remember.
Focusing on travincal runs isn't a bad idea, but it is tedious. I think from level 21-22 in normal you get some decent experience, but you'll need 50-100 runs to get the levels needed. If you make it to nightmare, and maybe start focusing on gold gear+charms and gold find, then you can start gambling, and thus make for a good ladder start.
Obedience is a good aim for a weapon, especially concentrate, i've used that in nm travi and it wrecks them. The enchant trigger helps a lot with attack rating. But i've also found 299% ed bonesnap from normal diablo (quest bugged runs) and that made nm a breeze, bosses crumbling super fast ^^ Taking huge portions of damage off when crushing blow triggers.
Barb makes for an interesting hardcore challenge. You are supposed to be tanky, but still have to be careful. One simply cannot engage if there are too many enemies ahead, one has to portion them up in manageable pieces and play tactical. And learn to use the R button frequently. And using the merc cleverly is paramount too. The big problem can be resistances. So making a high resistance shield is important, meaning you need a good 1-handed weapon, which can be difficult to find. Crushflange is an item that can drop early, if you find that, it can be a boss killer with the 33% crushing blow. I can imagine with a super fast double swing, maybe augmentet with frenzy, and high resist gear, should be a hilariously fast boss killing build. Can be upgraded too, damage still low, but useable.
D2:R