Battleborn

Unique Legendary: Hedronic Amplifier. +% Attack Damage, +% Skill Damage, Attacks charged by Hedronic Arc will slow on the first hit for each enemy.

Ascent from the Pits: Win a match on each Tempest map at least once (Echelon, “The Saboteur”, and “The Heliophage”). Self-explanatory. Maybe bring some friends to make sure the map you need gets voted (though obviously you can also do the two PvE missions solo).

Defender, You Have Become Defenseless: Deal 30,000 bonus damage to shields. Means damage to shields done by Hedronic Arc. A quick way is to use the Bulwark Minions during the early part of The Experiment (see above), just note that Hedronic Arc deals some damage over time, so try to leave it some 15% of its shields (just past the 'M' on the 'M2' in its name should be safe) or the DoT will bleed through onto its HP and may kill it early.

Leading the Charge: Activate skill with a fully-charged Hedronic Collector, 200 times. Easily enough done in normal play, just remember to use up the charges as you get them. You could also go to the farm spots on Algorithm or Void's Edge to easily get fodder to build up charges, but I doubt that is going to be significantly more efficient than just playing the game.

In His Face: Land 12 critical hits on Lothar Rendain with Charged Hook. Refers to the charged Secondary melee (the charge bar on the left, next to your HP), not Hedronic Charges, and the melee doesn't have to be charged all the way. Obviously shields don't take crits, so you this has to be done in the final phase of his fight. You're probably best off soloing the mission to reduce the health and amount of mobs significantly, then just take your time; with a PUG this is going to be relatively tough, so if you have to, bring your own team. Due to the way the hitboxes work, you want to aim a bit to the left of his face.

Rise of the Thrall: Knock up three enemies simultaneously, 25 times. You need to use a Fully Charged Hedronic Eruption for this, so it takes a little setup, but the AoE is large enough that it's not too bad. Just make sure the targets survive the initial impact, so this one may actually be slightly easier in a group. A good spot is The Renegade for groups of Thralls, and Void's Edge solo for endless Varelsi Skulks at the end.

Attikus' lores are fairly straight forward and A Navy of Ducks' guide is good but old and missing some helpful tips to make these easier

Ascent From the Pits: Win a match on 3 unique Tempest maps
You can't really get these PvP anymore unless you have a group. Battleborn's online community is officially 100% dead. No one is playing it anymore. Thankfully this can lore can all be done solo using 2 story missions and 1 ops mission.
the 2 story missions you can complete for this are:
Episode 7: The Saboteur
Episode 8: The Helophage
the 1 ops mission is:
Attikus and the Thrall Rebellion
NOTE: the ops mission is only available if you purchased it with platinum coins (the special ingame currency). Formerly you could just have all the additional ops missions and extra characters instantly unlocked if you bought the Digital Deluxe Edition version or upgrade. Unfortunately this has been delisted from the XBL marketplace. You may still be able to purchase dlc codes for platinum coins at gamestop. Not sure if those will actually work though.

Leading the Charge: activate skills with a fully charged hedronic collector 200 times.
level 5 mutation: Charge Efficiency
makes getting this lore super easy. this mutation makes it so you only ever use 1 charge instead of all 5. all you need to do is kill 1 enemy to max out your charges again allowing you to rapid fire your skills for this lore.

In His Face Land a critical hit on Rendain with Charged Hook
can only be done on the final boss of Episode 8 The Heliophage.
you must hit Rendain in the head with . The description says it needs to be fully charged (there's a meter on the bottom left of your HUD showing you the charge), but I've heard you just need to get a critical hit with this attack full charge or no.
This is not easy. The video Ducks linked in his comments section is wrong. The hit box for Rendain's head is very small and Attikus' dorky fists are too big. Getting a critical hit is very random. Ducks' video tells you to aim to the left of Rendain's face. That did not work for me at all. I got it after about the 10th time of hitting him with the crosshair directly on his face. Its frustrating, but you will eventually get it.
NOTE: Rendain's attacks will knock you off the stage to your death. You want to fight him on the right edge of the map from where you enter the level (there are stairs that go up to a flat metal looking floor)

the remaining 2 lores are easy

- keep damaging enemy shields
- activate Hedronic Eruption ( skill) with full hedronic charge and hit at least 3 enemies