In order to unlock all the animals for the director mode you will need to collect the Peyotes in the game, there are 27 Peyotes in the game and only 21 animals to unlock,
Solobrus22 has confirmed that you do not need to collect the 6 underwater Peyotes as you cannot use them in the director mode, they are numbers 6,12,13,21,22,23 of the below guide.
There is an excellent guide here http://www.gamesradar.com/gta-5-peyote-plant-location-guide/ https://www.trueachievements.com/xI+Xbox+One+Ix.htm has details on what the 22nd animal is, which is not needed for the achievement
*** Spoiler - click to reveal ***The 22nd animal in the list is 'Wild Animal' a.k.a. Bigfoot! You can only collect him after you've obtained 100% No, Bigfoot isn't needed for this achievement, like the underwater peyote plants.
You do not need to collect them all again if you've already collected themIn order to unlock an animal you will need to pick up the Peyote using

which will spawn you in the body of an animal. You can either play out your experience until death or hold

to end your experience, upon returning to Micheal/Franklin/Trevor the animal should unlock.
You might only have 21 out of 22 unlocked as Solobrus22 notes that's when their achievement unlocked.TangerineGamer estimates this takes around 3 hours.
Once the animals are unlocked enter story mode and press

and scroll right using

to the 'Rockstar Editor' menu and press

to enter. Choose 'Director Mode' from here and accept the launch.
Click actors and pick an animal from the animals menu.
Once the mode loads the achievement will pop, here's a video (
https://www.trueachievements.com/DeaDLyxCaNaDiAn.htm notes that it might not pop straight away, they had to also return to the story mode)
Loading up director mode with any animal will also unlock