I’m not sure if people are aware, but you do NOT need to have purchased Pendles to unlock this achievement. I played as him in two different rotation weeks in order to unlock this achievement.
For the Lore challenges, you can obtain all of them by playing against Bots in the Versus Private, Versus Maps - Classic. I specifically played Meltdown on Outskirts as it has a good section for getting the Smoke Bomb done but also found the Incursion maps useful for Battleborn kills.
For Loadout I equipped Attack Damage, Health Regen, and Cooldown.
The Smoke Bomb, 50,00 Damage, and 500 Damage challenges can all be worked on together, as can the Tentacle Slap and Miasma challenges.
On the Outskirts map there is a useful Accelerator that can be reached by heading towards your initial left-hand incinerator (the one your bots are trying to sacrifice themselves in). Once at the incinerator, there is a path/track to the left that the enemy bots will emerge from. Follow this round and you will see a pillar and an Accelerator. It is this location that will form the basis of the Smoke Bomb and 50,000 Damage solutions.
Use Smoke Bomb 100 Times While Under Any Crowd Control Effect
This is fairly easy from the location outlined above. Just step out from behind the pillar and the Accelerator will slow you down. Hit LB to drop your smoke bomb and head back behind the pillar.
Deal 50,000 Damage to Enemies From Behind
This is pretty self-explanatory but can be achieved whilst working on the Smoke Bomb challenge. When the enemy bots who are heading for your incinerator have passed the pillar you are hiding behind then follow them until you are re-cloaked and then attack using your primary attack (RT) and Injection (RB). Selecting the Level 3 – Backstab (RT) and Level 6 – Sweet Spot (RT) Helix options will also help get this quicker.
Deal 500 Damage after uncloaking, 100 times
Should be achievable when doing the above challenge, as long as you were cloaked before you attacked.
Tentacle Slap Enemy Battleborn 50 times
Tentacle Slap is Pendles “B” button. As soon as you see an enemy Battleborn hit B and you should have this in no time. Whilst hiding behind the pillar you will most likely have a few Battleborn pass through that you can slap but otherwise this can be worked on with the Miasma challenge below.
Kill 20 Battleborn while Miasma is active
Miasma is Pendles ultimate skill and is available once you get your in-game character to level 5. I found the best way to attack enemy Battleborn was to use Smoke Bomb (LB) when you first encounter an enemy, then Injection (RB), a bit of primary attack (RT) and then finish with Miasma (Y). I found the Incursion game type best for this as it tends to run for the full 30 minutes and allows you to get your level up. The bots do tend to run when their health is low but the smoke bomb can often keep them stationary long enough for a kill. Useful Helix upgrades are: Level 1: Poison Cloud (LT); Level 2: Flash Bomb (LT); Level 3: Backstab (RT); Level 4: Savage Cut (RT); Level 6: Sweet Spot (RT); Level 8: Acidic Blast (RT).