Vtyjkyuk

My trilogy features an otherworldly monstrosity called Fenrisúlfr as it's primary antagonist, whose characterisation takes cues from Nyarlathotep, Sauron, Showa!King Ghidorah, Sutekh 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.

Well, a start could be that you can make Fen occasionally do things that we lowly mortals perceive as 'good'. After all, incomprehensible does not always need to screw over a mortal, at least in the short term.

An example that I myself have written is the Rakh'vash. Consider him a mixture of Azathoth and Shuma-gorath, the Embodiment of Chaos. He speaks to select mortals called 'Chaos-speakers', and occasionally crafts flesh golems imbued with a part of his mind known as 'daemons'.

Anyway, as the God of Chaos, his ultimate aim is simple; make the universe overall more chaotic, both on the micro and macro scale. And while this seems like it could only be considered 'bad' from a mortal scale, consider this.

There was once a time in the world where chaos-speakers were killed at birth for their existence by a world-spanning empire. This same empire's dominant species considered themselves the only sentient race in existence, enslaving other very much sapient humanoids and repressing individual rights even among those they considered people.

It was an orderly society, to some a golden age. In fact, its higher-ups were comprised of Order-speakers, those who spoke with and carried out the will of the Rakh'vash's sworn enemy, Rakh'norv, the God of Order.

Now, the Rakh'vash had no concerns about human dignity and individual rights. It didn't care about the suffering and the enslavement and the utter dearth of soul in the world. All it cared about was that the Rakh'norv was stabilising a planet with sentients on it and this undermined its only purpose in life.

So when a chaos-speaker child next slipped through the world empire's genocide system undetected, he lent this young man his power. Slowly and surely, this man grew to be a revolutionary who toppled the empire and allowed other races to rise up in prominence. In the end, a more vibrant, individualistic, egalitarian (for its day) age came about...

...and it was all thanks to an eldritch abomination who ultimately couldn't care less about any of that. That's the thing with incomprehensibility. It doesn't look like it's aiming one way or another by human standards.

I think you could take some guidance from how Marvel represents Galactus And possibly even Thanos Who are both highly destructive beings of near infinite power.

They are not really represented in a great light, but the motivations they have are well beyond the usual good & evil type of scenario.

You might also want to consider The Watcher and how whilest he had immense power he chose not to act, with extreme consequences...

Eldritch horror is all about what you don't say about your villain, the less the reader knows the better. The point is to remind the reader that humans are tiny and irrelevant in the grand scheme of the universe. Usually you do this by invoking deep time and positing the idea of evolutionary and cultural dynasties that have existed throughout all of the history of the universe, pointing out that humans are Johnny Come Lately. Alternately you may point out that we live on a tiny speck in a huge and indifferent universe.

Either way eldritch horrors should barely notice humanity let alone think we're important enough to deliberately mess with us. When the human race gets whipped out it should be a side-effect not a goal, like someone being randomly hit by a wrecking ball because they're standing behind a wall to get out of the wind and light a cigarette at just the wrong moment, their body buried unnoticed in the rubble.