The first thing you need out of your mouse skills as a Barbarian is reliability. Your Primary skill should always be one of your mouse buttons, since you rely on it a lot- it’s your main source of damage, and the way you build your Fury to fund your other skills. Your other skill depends very heavily on which Primary skill you choose, though. A Barbarian who has Bash, unless it’s Clobber, is likely to have some issues with having to repeatedly catch up to their target as they send it reeling around the battlefield. For those Barbarians, Whirlwind Attack will allow catching up while still dealing out more damage, and a good alternative is Weapon Throw, which doesn’t care about the range and, combined with Bash, should be enough to kill most common enemies. Another good idea would be to pick an area attack- Rend, Cleave, Whirlwind, Revenge, or Overpower are good choices. Barbarians using Frenzy are generally free to pick whatever they like for their other mouse skill, but should probably invest in something like Ancient Spear, Stomp (Wrenching Smash), or Furious Charge to ensure that they can immediately get another target to replace whatever they just beat to death- preferably a skill that doesn’t cost a lot, or that has a rune to reduce its cost or cooldown. Barbarians who favor Cleave don’t need an area attack like a Bashing or Frenzy Barbarian might, and they don’t have to worry about retaining a speed buff from Frenzy. Because of this, they can favor Weapon Throw to give them a quick method at hand for smacking (and with its low cost, often repeatedly enough to kill it) things that are at a distance. Another alternative would be to rely even more heavily on Cleave and stick something with a modest cooldown or Fury cost and high damage in that spot, such as Overpower or Seismic Slam. Generally speaking, Barbarians want to make sure they have some degree of reach out of their mouse skills, and a Primary skill that works well with their playstyle.
When it comes to the hotbar, the Barbarian is going to be looking for a set of skills that don’t serve one another’s purposes- either they’re situational for different situations, or they are a stack of buffs that all benefit you in the same way. Most Barbarian skills with cooldowns have short enough cooldowns that they should be ready by the time you’re in a situation to use them again, so you shouldn’t be trying to overlap skills with similar purposes- Leap and Furious Charge, Weapon Throw and Ancient Spear, Revenge and Overpower. Typically, a Barbarian will want to make sure they have Threatening Shout, Battle Rage, or War Cry- Inspiring Presence users may want two of these, but all three is over-investing and may leave you short on situational skills. Ignore Pain is handy if you happen to lean towards Stomp (Wrenching Smash) because anything that survives past the stun is going to be trying to floor you. Most of your hotbar skills as a Barbarian should not be frequent-use and Fury-consuming at the same time- that’s a job for your non-Primary mouse skill. If you find that your mouse skills serve your purposes perfectly for damage dealing and you’re never looking for additional hits, stacking your hotbar with tools like Overpower, Revenge, Battle Rage and Wrath of the Berserker can build up that damage to astounding heights. Alternately, you could invest in Ignore Pain, Revenge, War Cry, and Stomp to put a huge crimp in the incoming damage.
The greatest effect Elective Mode has on passive Barbarian skills comes in with Inspiring Presence, Brawler, and Boon of Bul-Kathos. Inspiring Presence gets that much stronger when you’re not only maintaining War Cry but firing off either Threatening Shout or Battle Rage on a regular basis- 1% per second is just a nice extra when you have one shout, but when you keep shouting all the time, it counts for a lot. Brawler becomes very nice when you’ve got more ways to get enemies into proximity with you, and being able to add Wrenching Smash and Call of Arreat to each other on your hotbar lets you keep putting all the enemies in a large area next to you and maintaining Brawler’s bonus. Boon of Bul-Kathos is nice when you have only one Rage skill, but when you have all three it becomes pretty ridiculous- at a 90-second cooldown each, you’re triggering one of those every half a minute, which is very strong. If you’re operating with few cooldowns from your skills, then attack speed buffs are going to be a good friend of yours in equipment, particularly if you generate plenty of Fury. If your Fury costs and cooldowns run high because of your skill choices, on the other hand, you may want to invest more in Vitality and Thorns effects, just to stay more stable.