My trilogy features an otherworldly monstrosity called Fenrisúlfr as it's primary antagonist, whose characterisation takes cues from Nyarlathotep, Sauron, Showa!King Ghidorah, Sutekh, Makuta Teridax and the Gravemind. It is single-handedly responsible for every bad thing that happens in the series and the actions committed by several antagonists. Fenrisúlfr visits certain people in their dreams, as loved ones or religious deities to persuade them into finding ancient weapons called Divine Tools that form an interdimensional portal allowing Fenrisúlfr to escape from within a black hole, in exchange for granting their deepest desires. Throughout the series, Fenrisúlfr tries to goad the protagonist into gratifying his selfish desires and clinging to his nihilistic worldview, while appearing to him in the form of a monstrous dog.
Towards the end of the trilogy, it's revealed that Fenrisúlfr is responsible for the destruction of countless interstellar civilizations and aims to exterminate humanity out of hatred for organic life, which it likens to a plague that consumes everything in its wake. As such, it believes that the only way that it can prevent the universe's destruction is to eradicate all organic life in the universe. To this end, Fenrisúlfr took it upon itself to manipulate various historical and supposedly mythological figures in order to achieve its end goal of systematically destroying all life, while serving as the inspiration for monsters from various Indo-European and Afro-Asiatic Chaoskampf myths such as Tiamat, Kāliya, Ḫedammu, Leviathan, Apep, Typhon, Yamm, Illuyankas, Vritra, Aži Dahāka, Satan and, as its name suggests, Fenris.
Rather than writing Fenrisúlfr as a villain defined by it's desire to cause death and destruction, I want to portray it as a being that is simply so far beyond human comprehension that our concepts of good and evil cannot be applied to it.
How would I be able to achieve such a feat?
The problem here is that by giving him a clearly understandable (even if evil, misantropic) goal, you're making your Fenrisúlfr more human-like. Sure, we can say - by rough sketch - that it wants to eradicate life.
But to be truly "so far from human comprehension" we need to cut off any human understandable explanation from his actions.
Your question reminded me of Agent Smith's speech in the Matrix about humans being infestant and nocive as viruses: a memorable scene, and surely a villain-nesque one. Smith is a villain and can be surely defined as evil from a standard ethic-point of view.
On the other hand, most Eldritch abominations in Lovercraft's lore are able to make humans go mad just by existing. C'htulu and his playmates can make you mad and transform your brain into pudding by lifting a finger, but it's ever unclear if they do that intentionally or it's just a side effect. Cosmic horror is given, in the end, by the terrible randomness of it all, that challenges and questions our innate sense of cause-effect.
By contrast, check this question on Worldbuilding's stackexchange about a somewhat good-willed abomination which tries to avoid human going mad: link
So, the point that I'm trying to make is that we may as well synthesize Fenrisúlfr's goal as "wanting to destroy life", but we should stop at that. Why does it wants that? How does he plan to achieve that? Ultimately, the less we know, the more uncomprehensible he will be.
The reader (and the characters) should ask themselves: yes, it appears it's trying to destroy life, but are we certain of it? Sometimes Fenrisúlfr's influence should produce inexplicable effects, or straight up conter-intuitive ones (like giving power to a good, untainted character, or partake in the defeat of a human villain, or destroying a dictator in the midst of a killing spree...).
That's because the way Fenrisúlfr perceives reality should be so alien from our own that we cannot possibly comprehend it. So, it may as well try to manipulate humans, but it must be done in a so subtle and a twisted way that it's not straight-up recognizable.
In western culture, we have the idea of the tempting devil - Satan popping up from smoke, promising to grant us wishes if we do bad - but that's the very thing we're trying to avoid. Satan, in most portrayals, is simple: we undestand him roughly and man, he does understand humans a lot. So we can deal with him, and sometimes trick him into failure.
But Fenrisúlfr? Its mind is so complicated that we can't grasp but a small portion of it.
And, while being generally more powerful and intelligent than humans, there's a good chance that he can't comprehend us in the same way we can't comprehend ants. Our minds are substantially different. That may be a good reason why sometimes, Fenrisúlfr's plans fail: it just cannot comprehend fully human behaviour (after all, we're alien to it).
So, in short, my suggestion is: remember that when writing Fenrisúlfr, if you apply human-like desires or intentions to it, they should be only gross approximations. Make it random. Make it fail, if need be. Make it unforeseeable.