Battleborn

The Armor Costs More Than You: Deal 40,000 damage with Aerial Assault.
Aerial Assault is Caldarius' ultimate ability which means you need to get to level 5 in the match in order to use it. He boosts straight up into the air and then dive bombs to the ground, causing an explosion. BE AWARE, when you use his ult inside, if there isn't enough space above you, it will automatically cancel the ult, deal no damage, and require you to wait for a portion of the ability's cooldown. This challenge should come with time of just playing the character, but if you want to finish it quickly then play the story missions and use it in groups of enemies to maximize damage.

Soaring Reputation: Double Jump 1,600 times.
Double jump is a unique ability to Caldarius, it allows you to jump twice. Honestly this will take quite awhile to do naturally so you'll need to boost it. Save this for last, and while you're completing the rest of his challenges simply double jump the entire time. When you finish the rest of his challenges, you'll probably need to do a lot more double jumps so the best way is to simply load up a private story mission, stand at the start and just double jump until you finish it. It may not be fun but it's effective.

A Vision of Solitude: Kill 200 blinded enemies.
This is not nearly as easy as it sounds. One of Caldarius' abilities is a flash bang which does a small amount of damage and blinds enemies. What makes this annoying is that the blind effect only lasts for 1 second so you have a very small window to kill the enemies. This should be done in story mode because of the sheer amount of enemies you face in campaign as opposed to versus, where you might not even get the kill. The trick is to get the enemies to very low health so you can kill them in that 1 second blind, but not low enough that the flashbang kills them. There is one very helpful helix upgrade, its the left upgrade on lvl 4 (flash barrage). This upgrade makes your flash bangs fire 3 smaller grenades in succession. This allows you to kill the enemies with the second or third grenades while they're still blinded from the first. Other upgrades that help are Rapid Dominance on the right of lvl 6 (reduces cooldown) and Brightblaster on the right of lvl 9 (increase area of affect). To save time, combine this challenge with the next one, for killing Jennerit enemies by playing either The Renegade or Heliophage missions.

The Arena and You: Partners in Freedom: Kill 500 Jennerit Enemies
This should not, under any circumstances be done in versus because it will take forever since the only Jennerit in versus are other players and the thrall mercenaries. This should be done on the Renegade or Heliophage story missions since it is comprised of almost completely Jennerit enemies. Try killing them while blinded to kill 2 birds with 1 stone.

Three-to-One Contact: Kill 150 enemies with Energy Blade while on a team with Rath.
Energy Blade is Caldarius' attack and is a melee attack. Rath isn't an uncommon character in versus so you can focus on minions to get the required 150. The easier and faster way to do it is to connect a second controller, start a splitscreen story mission, make the second controller Rath and then go to town on enemies with the energy blade and this challenge will be done in no time.