WARNING: Starting around June 2019, it appears this achievement is not unlocking when it should. I've contacted CYANIDE Studios and Focus Entertainment about this, though I'm not sure if I'll get an answer since their Technical Support page for this game is no longer in service. Fortunately, we've seemed to have sussed out a way to ensure the achievement unlocks, see updated solution below.
UPDATED SOLUTION (Last Update: March 2020)
To gain this achievement, not only must you respond in R'lyehian all nine times that you can do so, you must also do the following:
1.) Lower your SAN score as much as possible (down to psychotic) by end of game, especially by Chapter 9 (i.e., via reading all the unholy books, getting the hallucination in the sewer in Chapter 2, etc). This will get you the nine opportunities to speak R'lyehian.
2.) Starting from Chapter 9 and on to Chapter 14, make sure to speak R'lyehian in the nine opportunities listed below.
3.) At the beginning of Chapter 11, select both "Submit" and "Accept the Knowledge" during your conversation with the Leviathan.
4.) At the whaling station hallucination sequence in Chapter 13, "submit" all three times, (aka taking the treatment, eating the meat, and taking the gun). Low sanity characters may be forced to take these options no matter what they pick, but better safe than sorry.
5.) At the very end of the game (Chapter 14), select "Perform the Ritual"
Here are the nine times you need to speak R'lyehian, and the chapter they can be found in:
Chapter 9: Sarah Hawkins -- when you first find Sarah in her cell at the institute, you will get this option here. NOTE: If you do not get the option to talk in R'lyehian here you sanity score may be too high to complete this achievement.
Chapter 9: Officer Bradley -- MISSABLE Bradley is located on the second floor of the library at the tables once you return with Sarah to the mansion. Easy to miss him if you're not paying attention.
Chapter 9: Sarah Hawkins -- Once you go downstairs in the library after the Bradley conversation, you get pulled into a conversation with Algernon Drake and Sarah Hawkins. Once Drake leaves, you will talk to Sarah alone. The option is here.
Chapter 9: Algernon Drake -- This option triggers after talking to Sarah where the next objective is to talk to Algernon. NOTE: you have two opportunities to get this, one right before receiving the drawing of the bust and one right after giving Algernon the amulet. If you don't have enough SAN drop to get it the option the first time, finding the amulet will drop your SAN one last time -- so don't despair if the option isn't available when you first see it, as you'll have one last opportunity.
Chapter 11: Leviathan -- This option triggers once you speak to Leviathan again at the beginning of Chapter 11. Hilariously, this is supremely unmissable, as you have six R'lyehian options to choose from. All of them count toward the achievement, so pick what you like.
Chapter 14: "Marie Colden" -- There will be three opportunities to talk in R'lyehian in the same conversation when talking to "Marie" in chapter 14.
Chapter 14: Sarah Hawkins -- Right at the end of the game, when talking to Sarah, your last opportunity is here.
Complete the five tasks above, make sure you speak R'lyehian at all nine opportunities, then select "I will perform the ritual" as you last option, and the achievement should pop during the final cut scene.
(Please let us know in the notes if this method has worked for you -- we're been working on this via trial and error for a few months, and would appreciate confirmation that the above works to unlock the achievement now)