A lot of barbarians are having issues with Inferno difficulty, because the damage scaling is so punishing to melee characters. This is happening because Inferno damage is designed around having high resists, which a lot of people don't have yet. The truth is that barbarians are the most gear dependent class in the game, so things will get easier as more gear becomes available.
In the meantime, I've spent a considerable amount of time figuring out how to farm up my own gear. From all the approaches that I've tried, by far the most successful is a tank barbarian with a lot of defensive skills. For a given level of gear, I find that using my tank build I can farm elite packs a full act above what I could using a dps build. This sucks, because it pigeonholes the barbarian into a specific build. However, I expect things to get better a month or so from now, when a lot of players will have enough gear to farm inferno act 4 and whimsyshire. Once it becomes a contest of speed rather than survivability, barbarian builds should diversify again.
This build is very versatile. It can kill most elite packs, all bosses, and skip through zones. It's designed to get 5 stacks of Nephalem Valor, reach bosses and then kill them for the bonus loot.
Frenzy(Sidearm): I brought this skill because I wanted to have good single target damage for boss encounters. The Sidearm rune was chosen because it has the highest single target dps (considerably higher than Maniac in this build), while simultaneously providing a little bit of aoe damage, since the thrown axes can hit multiple enemies.
Revenge(Provocation): This skill provides the staying power when you're fighting groups of enemies. I chose the Provocation rune to make the heal consistent, which is important because this build depends on it to stay alive.
Ignore Pain(Iron Hide): An amazing panic button, it has saved my life more times than I can count. It has a decent duration and short enough cooldown that I can depend on it to get me out of tricky situations, such as getting frozen on top of aoe damage. Also, for especially challenging encounters, you can dps while it's up and kite while it refreshes.
Leap/Iron Impact: Leap is a fantastic skill. It gives the build much needed mobility since you'll often be blocked by groups of enemies. Furthermore, the Iron Impact rune allows you to rotate defensive cooldowns (Leap/Ignore Pain/Leap). With my gear, the 300% armor increase means I take 60% less damage!
War Cry(Impunity): This is here mostly for the Impunity rune, since I stack resists and adding 50% to that is very effective. With my current gear, having this shout up reduces incoming damage by 33%!
Furious Charge(Dreadnought): This skill has two purposes. First, it complements the mobility provided by Leap, and if you use both in succession it is possible to deaggro packs that are chasing you. The second purpose is for the heal, which can be used as a backup if you get unlucky with Revenge, and also for miscellaneous healing when you are trying to kite and not enough enemies are around you for a Revenge heal.
Passives: Nerves of Steel, Superstition, Berserker Rage.
I chose two defensive passives to further reduce incoming damage. I chose superstition over Tough as Nails (12.5% reduced damage taken) because I'm not as concerned about physical damage as I am about elemental. Berserker Rage is almost a flat 25% damage increase, as you're often at full rage.
The idea of this build is to maximize effective health, and then take advantage of the percentage based heals from Revenge and Furious Charge becoming large compared to the incoming hits.
When fighting challenging enemies relative to your gear, I find it most effective to do damage while rotating Leap/Ignore Pain/Leap and then kiting with Furious charge until I can do it again.
I've made a series of videos showcasing the gameplay and my farming route for each Act:
Act 1: http://www.youtube.com/watch?v=H9U8YkFJLb0
Act 2: http://www.youtube.com/watch?v=3Ygb4NS4JXw
Act 3: http://www.youtube.com/watch?v=w_yi1k6X8fQ (post T-nerf)
Act 4: http://www.youtube.com/watch?v=XSxwug0mkdg
I've also used this build to solo all of act 4 while rolling 5 stacks of valor, it took me about an hour.
The downside of this build is that the dps is very low. I make up for this whenever I can by farming with npc assistance, but I expect to be able to increase the offensive aspect as I get more gear. Once I can kill act 4 elite packs without dying too much, I'll transition into offensive stats while keeping survivability constant.
Resist All / Vitality (Try to get both on all pieces)
Life on hit (in the hundreds)
Physical or Fire Resistance (Desecrate, Molten, Fire Chains, Mortar are all fire and problematic)
Life regeneration (Always nice for when you're kiting, especially during boss fights)
Furthermore it's important to maximize your block chance. I currently have 47% with a Sacred Shield, Justice Lantern, and Helm of Command, but I realize that not everyone will have access to that. Block % is the most important statistic on your shield and often overlooked. Also, try to have at least 3k blocked amount, which can outright negate most hits if you have high armor and resists.
Other desireable legendaries are the String of Ears belt to reduce melee damage taken (as high percentage as you can afford each point is better than the previous one), the Band of Fallen heroes is a nice ring with Resist and Vitality, the Blackthorne Medal is an amulet with attack speed and life on hit (2 set gives 100 vit) and Immortal King 2 set has a bonus of 60 resist all.
Bring the Enchantress companion for her Powered Armor buff, it makes a huge difference. I made a video going over how I spec and gear her, check it out here:
Hope this helps! Feel free to ask questions
EDIT: I wanted to mention a trick I discovered:
By using Furious Charge or Leap at the correct time, it's possible to avoid getting frozen. You still take damage, but it can be a lifesaver.