Battleborn

Sustained Across Time: Deal 10,000 melee damage in a single match or mission, 10 times.
Betrayal, Betrayer: Kill 500 Jennerit.
Render the Snake Headless: Deal 2,000 damage to Rendain with Catalytic Smash.
Axiom, Praxis, and Precept: Hit 3 enemies with Crossblade, 100 times.
Thank You Letter: Play 5 matches on the same team as Caldarius.

Sustained Across Time is pretty easy and straightfoward. I found that I was getting at least 10,000 melee damage in every story mission I did without having to go out of my way for more kills or damage.

Betrayal, Betrayer is pretty easy as well. I got it easily just by playing through the story. Jennerit enemies are in most, if not all of the missions.

Render the Snake Headless can be done within the Heliophage mission. Catalytic smash can be used with . Just use it as much as you can against Rendain, and you should be able to get this within two runs.

Axiom, Praxis, and Precept will probably be the last one you need to get done. I had about 70-80 when I finished all of the other ones. Crossblade can be used with . You should use the second level helix upgrade, Anger's Echo, which makes another Crossblade that returns to you. You should also get the level 8 helix upgrade, Energetic Projection, which doubles the range of Crossblade. I found that a good level to grind it on is The Renegade at the end when there are a lot of enemies spawning on the platforms outside.

Thank You Letter can be a bit of a pain, but if you have a friend that has Caldarius unlocked, it can be done fairly easily. If you do this in story, you can just waste all the extra lives by jumping off a cliff, and if you do this in versus, you just need to finish the match. You don't need to win the versus match or complete the story mission