You must catch a fish with your hook, then release it I think with right-click while above sea level. Diving is slow and the fish will swim away. It's easier when you position yourself beneath a fish a quickly rise to catch it. The hook needs to go over the fish, not just your ship.
A bigger, slowly moving vessel named the Rooster Vessel will apperar after a few minutes in the game. For me it was always spawned fairly high in the sky, so keep looking for it. You can barely miss it, since it's the biggest and most complex ship in the sky. And it has a rooster on top, hence the name.
I think this one triggers when you kill enough ships to spawn the Rooster Vessel and just see it on the screen. It might not be a requirement to destroy it yet.
To the right of the palace there is an area where a huge fish swims around. It has a large mouth and can swallow you. This doesn't kill you, but gives you control of the fish. All you need is to swim around the area (you can't go round the map and don't earn any points) and swallow the smaller fish which get spawned regularly.
When you destroy the Rooster Vessel a star is dropped. All you have to do is pick it up after it lands on the bottom of the sea.
Duh! Play well enough and destroy things for about 5 minutes.
So you're valiant if you destroy a small town of pilots... OK, I guess. This achievement may require some skill. The X2 perk helps here.
You should pick up the 'dropping ballast while ascending' which shoots your weapon as you climb. It allows you to climb faster before your air is depleted, but the 'underwater endurance' perk can help too.
The airships are larger ships that fire a stream of cannon balls at you. It looks similar to the achievement's icon - big balloon with a cannon that follows your movement.
These get spawned at sea level. They are more difficult to kill than the smaller ships, but getting 3 kills shouldn't be a challenge if you focus on them.
Honorary member of the cannon club
Self explanatory. This one gets enabled even if you don't focus on it and play well enough to survive about 5 minutes while destroying everything around you.
Chairman of the cannon club
Above +4. Are there more than 12 I wonder? I think you need the Rooster Vessel.
Your hook can pick up the barrels left behind by destroying the smaller vessels floating at sea level. Use your weapon to kill them, though. Afte you catch it, right-clicking to destroy the barrel and any enemies and their projectiles on the screen.
Keep going with the above achievement.
Just keep playing without sucking too much. It may be linked only to the ships on the water, though.
Love, honor, valor
This one doesn't need a single run, it's cumulative. So you can safely suck, if you still manage a kill or two on each run.
Pile of debris
Easiest achievement by far. You have to die. EZ.
Keep going left or right for a couple of minutes, ignoring enemies, until you get this achievement.
You might think, wrongfully, that there's enough time while falling from the highest spot where you can shoot to destroy 15 enemies before you reach the water. There isn't. Instead land on a body of land and shoot away.
The palace near the spawning point has a moon on top of one of the towers. You have to land inside it and wait a few seconds.
In the area where you find the big fish, the one that can swallow you, there is a deep dive spot. There are enough bubble healers to keep you alive. After collecting it with your hook you have to climb out of the water.
Stranger and Jellyfish
Pretty far to the right of the big fish, between some large mountains and after the spot where you can see a sunken ship, there's a dive spot that has a huge skeleton near the top. On the bottom of the chasm you can pick up an urn or ceramic-looking vessel. When you bring it out of the water it eventually starts to release something, probably fire, and you can't shoot any more. You have to return to the water where you will turn into a jellyfish to get the achievement (white creature with tentacles). To find the Stranger you have to dive to the deepest spot which I think is further to the right and keep going downwards until the screen goes darker. The game will respawn you above water.
Doctor on board
Keep playing until you get to level 10. This is cumulative, so it's not on a single run.
Get the full crew, but I need help to find all of them.