The following achievements will unlock automatically as you progress through the DLC's main story, with the exception of "Good One, Babe," as you will need to meet certain requirements in order to obtain it.

Party's Still On!
Find the party.

Once you've arrived at the Lodge for the first time, talk to Hammerlock to conclude "The Party Out of Space" mission and you'll unlock the achievement.

Older Than The Trees
Kill the Wendigo.

At one point in the story, Hammerlock will accompany you on a Wendigo hunt. Defeat the boss, pick up its parts and talk to Hammerlock to conclude "The Horror in the Woods" mission.

Love Is The Fifth Dimension
Marry your friends.

After beating the final boss, talk to Hammerlock and Wainwright to wrap up "The Call of Gythian." Once the short cut-scene of the two getting married plays out, the credits roll and the achievement should pop up.

Good One, Babe
Watch Deathtrap kill 50 enemies.

Right at the start of the DLC ("The Party Out of Space" mission), you'll meet Gaige for the first time where you'll be tasked with fending off a small wave of enemies. Instead of getting involved, let Deathtrap run amok and get as many kills as possible (Gaige getting kills also counts towards this achievement, although Deathtrap is generally more powerful and will almost always be the one to get the kill). You can speed up the process and wound nearby enemies, but Deathtrap or Gaige need to be the one to finish them off. If you're downed in all the chaos, both of them will come and resurrect you. There are two more missions, near the end of "On the Mountain of Mayhem" and at the start of "The Call of Gythian", where Deathtrap gets involved and, in both instances, can one-shot nearby enemies.

If you've completed the story and missed this achievement, you can go back to Xenocardiac Containment on Negul Nashai (the last, open area on the ship) and redo the "Empowered Grawn" boss battle. Deathtrap will reappear as part of the boss fight and you'll be able to farm Grawn's minions for the achievement. If Deathtrap happens to only go for the boss, you'll have to draw in nearby enemies towards him so that he can shift his focus onto them. To restart the fight, simply exit to main menu, resume right afterwards and go back to the boss arena.

Alternatively, you can either use one of your other characters or create a new Vault Hunter (all of your progress will be stored with the previous character that you can access later) by going to main menu and under "Load Character", select new game and choose "Guns, Love, and Tentacles" to start right from the beginning of the DLC. Whichever of the two options you've chosen, rush to the starting point where you meet Gaige and let Deathtrap kill your enemies.

A less common way of achieving it is to join a friend's game and play from wherever they left off, but this requires them to be at a very specific point in the story and other methods are way more efficient.

NOTE: Before Deathtrap kills everyone, quit to main menu and then go back to the story. You will restart the fight, allowing you to quickly farm Deathtrap kills. All of them count towards the achievement progress and kills from your other profiles also carry over.

Additionally, if you happen to have the Night Flyer gun, you can use it to bring all your enemies down to 1 health and speed up the process.

Industrious In The Face Of Cosmic Terror
Complete all Crew Challenges and Side Missions on Xylourgos.

All crew challenges and side missions will become available once you've completed the main story.

Skittermaw Basin

• Call of the deep
  
• We Slass!
  
• We Slass! (Part 2)
  
• We Slass! (Part 3)

The Lodge

• The Propritor: Empty Bottles
  
• Sinister Sounds
  
• The Nibblenomicon
  
• Happily Ever After (unlocks after finishing the story)

Cursehaven

• Cold Case - Buried Questions
  
• Cold Case - Restless Memories
  
• The Proprietor: Rare Vintage

The Cankerwood

• The Great Escape part 2
  
• Cold Case - Forgotten Answers

Negul Nashai

• The madness beneath

Skittermaw Basin
Cursehaven
Dustbound Archives
The Cankerwood
Negul Neshai
Heart's Desire

The following achievements are independent from all missions and can be completed at any given point.

Whispers On The Wind
Collect all ECHO logs on Xylourgos.

This is best showcased with a video, as it's hard to give you their precise locations on the map with just screenshots. There are a total of 18 ECHO logs. Credit for the video guide goes to Indecisive Prophecy.

A Hunter, Through And Through
Find and kill all four rare spawns.

The achievement's description is sort of misleading, as they're not that rare. Takes somewhere between 4-10 tries to spawn them. They tend to stand out from the rest of the enemies and have additional health bars. If you fail to spawn them, simply exit to main menu, resume and go back to your desired location. Repeat this process till you get them to spawn. Keep in mind that this achievement might glitch out at times, as several people have reported issues with progressing past 50%.

Cursehaven - Amach
The Cankerwood - Fungal Gorger
This section of the map is devoid of enemies. Gorger will almost always spawn from the back end.
Negul Neshai - Shiverous the Unscathed
Shiverous always flies in from the far right side. You'll need to stand near the cliff to spawn the dragon.
Negul Neshai - Voltborn

I'm Floudering Up Here
Listen to 25 fish jokes.

Once you get to the Lodge, you can find a fish trophy mounted on a wall to the right of the bar. You need to interact with it 25 times in total to earn this achievement. Although there are a few more fish trophies around the world, this one's right next to your point of interest and is easy to farm. When you're done interacting with it (make sure the fish spits out the loot for it to count), quit to main menu, go back to the Lodge, listen to another fish joke and repeat this process till the achievement pops up. Keep in mind that the Steam counter is either bugged or delayed and does not accurately reflect your progress.

Alternatively, if you have the "Talking Fish" decoration, you can place it in your room on Sanctuary and farm the jokes in one go. I believe the trophy itself is a random drop and can usually be found inside fish nets all around the DLC world (there are some around the Lodge as well). Getting one is based on luck, but the drops aren't that uncommon and you should also double check to see if you happen to have one in your inventory (I got mine without even realizing it). Credit to NightRaven for telling me about this.
A snapshot of the fish's location.

Hope this helped you out. Feel free to give feedback or correct me if I've made any mistakes.