Battleborn

Bite-Sized Wonders: Deal 80,000 damage with Chomp.

Pretty simple; not a lot to say about this challenge. There are a few helix upgrades which increase Kelvin's Chomp damage:
1) Level 4 Slow Food (increase damage to slowed enemies. Note that you will need to take the level 3 upgrade Icemaker if you wish to be slowing the enemies yourself)
2) Level 9 Sawtooth (+15% health damage)
3) Level 2 Mutation Diffusion (Character Rank 5. Chomp deals extra damage equal to 3% of his current health. Pair this mutation with gear to increase Kelvin's max health for increased damage bonus)

I got this playing casually in Meltdown, but farming missions with waves of enemies would work well. Taking Kelvin's Overeater upgrade at level 9 to decrease Chomps cooldown is a good way to deal more damage.

That's an Ice Shield: Absorb 50,000 shield damage.

Every time Kelvin uses a skill Kelvin will gain a temporary shield. There are three Helix upgrades which increase the amount of shield Kelvin receives to allow you to take more shield damage:
1) Level 2 Strong Wind (+50% shield strength)
2) Level 5 Ice VI (+20% shield strength)
3) Level 4 Mutation Shield Snacker (Character Rank 7. 30% of damage dealt with Chomp turns into shield)

It is easy to do this in missions where waves of minions spawn. Simply spam Chomp facing away from enemies and let them take out your shield. Using cooldown gear and taking the level 9 skill Overeater will allow you to use Chomp more frequently for even more shield.

Miko Are All My Friends: Play 5 matches on the same team as Miko.

Easiest way to do this is to grab a friend/second controller to get both characters together and load up a mission on Hardcore. Simply jump off an edge to 'complete' the mission. Repeat 5 times and the challenge is yours. The first mission The Algorithm is a fine place to do this for anyone who hasn't played any story missions.

The Best Defense is De Fence: Stun 100 enemies with Ice Wall.

Ice Wall is Kelvin's Ult which is gained at level 5.
Again, missions which spawn waves of enemies are the easiest places to do this. While tedious to wait 60 seconds (or get cooldown gear) every time to use the skill, it isn't particularly hard to do this. Try to place the wall on a group of enemies and make sure the wall is spawned on top of the enemies to actually stun them. Just remember that Kelvin's Ult has a 1 second delay.

Micro Macro Mist Me: Stun 3 enemy Battleborn with Sublimate simultaneously, 25 times.

This is the 'hardest' of Kelvin's challenges. I found the easiest way to get this was to play on Incursion as players always group up at certain choke points on each map. Wait until your team is backed up to your Sentry or the enemy is backed up to theirs and then Sublimate in and out quickly. I 'lucked out' and played against a 5-man premade which always stuck together. It was pretty easy to stun multiple enemies at once.
Note that sometimes it is hard to see if you actually stunned the enemy, and sometimes it may look like you pass right through them and yet they don't get stunned.
This is the only lore challenge with Kelvin that you need any real luck with, and so it may take the longest to achieve.