Welcome! Anyone who's read the works of H. P. Lovecraft knows about Cthulhu, and most people who haven't read his stories have at least heard of him and could probably recognize a picture by the dragon wings and tentacled head. If you've read the books, you probably also know the sentence "Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl ftagn", which translates to "In his house at R'lyeh, dead Cthulhu waits dreaming." What you definitely don't know, because Lovecraft himself never elaborated, is how to pronounce it or how it was translated. There are fewer words in the original than in English, and "Cthulhu" and "R'lyeh" appear in the wrong order, so there has to be some non-English (or non-human) grammar at work here, and that's assuming you can pronounce it in the first place.
I'd like to preface everything with the disclaimer that I am not myself a linguist. I have never formally studied linguistics at Miskatonic University or even Massachusetts State. However, I take a very strong interest in the structure of languages, have mostly taught myself linguistics on the internet by absorbing information here and there, and while I definitely don't consider myself a professional, I think I can safely say I'm much more so than the average person. And with that background, I thought it would be an interesting idea to expand the language of Cthulhu from that one sentence to an entire, functional language that could actually be used. You might ask me why. That is a very good question and I have no idea whatsoever. I don't expect that anybody will learn to speak it because why would they? I suppose some people might; people learn Klingon and speak it to each other. Some people learn Elvish. But the director and author respectively probably never expected that. I can only assume, with the world as big as it is, somebody, somewhere on the internet, will probably find some value in the whole thing.
Although I have another invented language up on my site here, Ebören, I expect this one will differ significantly in style. I don't know until I try it, but in general I expect Cthulhu's language to be described in an altogether more serious tone. Part of that is the subject matter; to make jokes about the Old Ones and the Elder Gods would be sacrilege, not against Cthulhu (though of course that too) but against Lovecraft himself and the spirit of his writing. Furthermore, Ebören, while it may not be very similar to English, is at least designed to be learnable. Any language spoken by Cthulhu is going to be fairly inhuman. So there will probably need to be more careful detail on the exact nuances of pronunciation and grammar, because all the things we do instinctively while speaking English are just arbitrary English-centric and human-centric conventions. There's no rule that Cthulhu has to operate the same way. In Ebören I can just say "pronounce this like the s in snake"; here there may be more complicated sounds that don't exist in English. Of course, it isn't that bad; if a hundred ritualists can learn to chant it under high pressure, obviously it must be pronounceable, no matter how it looks. It is something that has precedent in human language. But it isn't necessarily how you might expect things to work. You'll see.
There are three steps to creating any language. There's phonology, which is just Greek for sound; this is where you decide how your language will be pronounced, how you'll write it if you're using writing that can be pronounced (as opposed to Chinese which can't), and which sounds if any will exist that aren't in English, such as a rolled r. In English, "drick" isn't a word, but it could be. On the other hand, "mvasdra" just isn't possible; you can't pronounce it. However, it would be a valid word in Russian. On the flip side, "strengths" is valid English but not Chinese. That's phonology. The second major step is grammar, which is where you decide how words relate to each other, where you'll have things like verb tense or plural nouns. And finally you have vocabulary, which is when you actually create words for different things. At the time of this writing, I have a pretty solid idea for phonology and a basic starting point for grammar. I don't know when or if I'll get to the point of completing all three steps, but I can at least set down what I have, and come back to it whenever I have more material.