Finish a run by passing the rainbow line at 6000m depth. It's not required to do anything else. The boss is at around 3500m but is easy to avoid. If you beat both stages of the boss, he drops the Abyss Core, which you can take back to the vessel at 0m to win the run.
If all achievements worked on same depot, it would be doable in around 90 minutes. The developer said there will be no more major updates.
Achievement overview
Achievements are mostly self-explanatory except that "return" now means crossing the "final wave" at the bottom.
Don't skip the tutorial, there is an achievement for it.
Have all of the starting equipment fully upgraded to +13 and equipped, and have the buffs activate a few times.
Finish a run with the following conditions (some can be combined): holding the core, after finding three chests, with 1000 coins, without taking damage, without catching anything, with almost no oxygen, after a Blood Frenzy and taking damage, with Abyss Gaze (I think this is when the abyss bar is maxed). Also finish a run by taking the core back to the ship.
Gameplay tips
Upgrades and items persist between runs (be sure to choose Continue on the main menu), but if you die you lose half your coins. Restoring air at upgrade stations costs a few coins so make sure you have some on hand.
Item upgrades cap at +13, you will want to do this as soon as possible to make everything else easier, it doesn't take long.
I never got equipment drops, just rings from enemies, but some of them have very useful abilities. At max level they grant an extra bonus, which stack. Each maxed Longline Ring gives +1 harpoon when shooting.
I'm not sure what all the equipment abilities do, but the maxed Harpoon Rig slows time while aiming, which can be very useful during the boss fight.
The first phase of the boss can be easily killed with an upgraded harpoon, but the second phase is pretty difficult. I'm not sure what damages it exactly, but the harpoon does almost nothing. The best strategy I found was to run away as soon as you can move, know where 2-3 upgrade stations are, and move between them while dodging the boss and killing minor enemies which will start swarming you. Sometimes the boss can get sort of stuck right on top of you and it's difficult to escape, so it might take a few tries and some luck. Keep moving around, but be aware of his location when he's almost dead so you can pick up the core.
Broken achievements
Need to find a depot for these or find out if they don't unlock how I expected. Game files seem to be encrypted and can't be decompiled.
Dragon's Remnant - Defeat the Abyss Dragon normally
Dragon Trophy Hunter - Defeat the Abyss Dragon normally 3 times
First Light Home - Complete your first regular dive successfully
Low-Oxygen Return - Return with 10% oxygen or less
Three Safe Returns - Complete three regular dives in a row
The Abyss Does Not Wait - Let a dropped Abyss Core expire
Also can't confirm if the first batch of achievements still work on the current depot. Catching 500 fish will probably come naturally but won't take very long to finish anyway.
Be sure to back up your save before depoting, which I believe is located at (may differ based on Steam installation):
C:\Program Files (x86)\Steam\userdata\(Steam ID)\4777880\remote\savegame.json