There are multiple achievements for different endings. You will need to play through the whole game at least four times to get them all since the developer(s) couldn't be bothered to add a chapter select.
Complete Achievement Guide
Hit a bull's eye with an axe from at least 10 meters.
This can be done in the castle garden room. Pick up one of the hatchets lying around and throw it into the center of the target. It is currently bugged, which means you can throw the axe into the center while standing directly in front of it if you really don't want to try.
No One Behind The Camera
A vintage selfie with both heads in front of the camera.
Once you make it past the corridor of death and reach the room with the sundial, you will see a camera in the corner. Walk up to the camera as if you are about to take a picture (line up your cursor with the red button on the right side). Next, stay still and don't move the mouse. Have your friend grab your head and place it on one of the chairs. Click the mouse to interact with the camera and take the picture.
A Generous Companion
Serve a nice and fresh beverage to your favorite companion.
In the garden, go to the table underneath the boar head with the mug on its tusk. You should see a small keg on the table. In order for this achievement to work, you'll need to drop a mug of ale onto your own head.
Grab a mug and interact with the keg to fill it with ale. Have your friend walk up next to you and grab your head and immediately drop the head directly in front of you. Now, drop the full mug onto your head.
Let's Call This a Necessary Sacrifice & One Out, One More To Get Out
In the final room with the rocket, once the launch sequence has been initiated, you will have 15 seconds before the rocket launches. One person will need to enter the rocket and fly away with it. This person will achieve the One Out, One More To Get Out achievement.
The other must stay in the control room (the room where the rocket launch button is) and watch the rocket take flight. This person will achieve the Let's Call This a Necessary Sacrifice achievement.
In order to get both achievements, you will have to play through the entire game twice, since there is no chapter select option.
Two Heads No Second Thoughts
Launch the rocket in the final room without anyone flying away with it. This is going to be extremely difficult as it is very tempting to get inside the rocket and fly away.
Two Headed Friendship
Escape in the rocket with your head on your shoulders while carrying your friend's head in your hands. We achieved this by having the person who pressed the button to start the 15 second countdown throw their head to the other player on the drawbridge leading to the rocket, since there is not enough time to run from the button to the rocket.
This might require some practice, since you only have one attempt to get the achievement or else you will have to start the entire game over.
The person who pressed the button should jump up on the railing of the balcony, detach their head, then jump off the balcony towards the other player. While you're in the air, throw your head to the other player on the bridge. The player on the bridge will have to catch your head then run inside the rocket.
Again, this will take quite a bit of practice, so make sure you can consistently throw / catch the head before you start the thruster ignition process. The player on the drawbridge should also be prepared to jump and catch the head just in case it is a little too high.