Eventually during your stay at the pool, either one of your own ducks, or the Key Duck behind the door on the second layer will interact with it, causing the Museum area to become available to any duck that makes it to the second floor via the geyser to the left of the pool you start in.
After this, over at the back of the unlocked Museum area is a hole:
All you have to do is wait for one of the ducklings to enter the Museum and go through the crack and into the hidden pool and the achievement is yours!

If a large duck happens to get stuck in the crack, you will unfortunately have to exit and restart.

Once the Key Duck has been unleashed from the Museum, the next achievement he'll be helpful with is "Splash!" -- This achievement relies on the Key Duck interacting with this barred-off grate that's close to the Museum door:
(version shown is the unlocked version)
Once unlocked, all that a duck has to do is swim into the bar-less grate, and they'll be teleported to another secret area of the map and the achievement will unlock.

Alternatively, you got it five minutes ago with a grate that's still locked, and have no idea how you managed this - this is thanks to the geyser, which has a high likelihood of launching your ducks far enough that they'll fall into the secret area and unlock the achievement that way.

... Or a duckling squeezes past the bars, as someone mentioned in the comments- this achievement's various possible unlocks are truly entertaining.

This achievement is rather straightforward, however may take some time to unlock.
All that has to be done is to get both the Stone Duck and Marble Duck to spawn in the pool, and after some time, both ducks will end up inside the display case next to the Museum entrance.
Once both ducks are inside the case, simply interact with it and the final achievement of the DLC will be obtained!