It is no surprise that trying to play Mythos monsters - non-human, inhuman entities that are terrifying just to behold, let alone comprehend - is not easy. As a Keeper, hands down the biggest problem you'll face in Call of Cthulhu is trying to portray and act as characters that aren't human beings. Protip - get real damn used to that, because if you're not able to engage in a bit of xenofictive musing on how your concept of say, Nyarlathotep acts, you are not at all prepared to be a Keeper. Turn in your EoD robes and staff now, wipe off your blood face pant, and go home.
It's important to note that here, when I say "monsters", I mean any Mythos entity - Hunting Horrors to Hastur; Zoogs to Zoth-Ommog. Whether Outer God, GOO, Mythos race, or Mythos species, it's all the same concept - they are antagonistic, but not necessarily villainous or evil/bad, beings that are responsible for causing the current struggle your Investigators face. It is therefore better to consider them like hurdles or beings opposite in motive to your Investigators. These hurdles can be dangerous, or completely mundane, but they should always be remembered first as characters with certain motives, not as just monsters for your players to kill. Call of Cthulhu is not a game of monster-slaying (unless you design your scenario around that idea, or are playing Pulp Cthulhu!); it's a game of investigation and wonder and awe and fear. Treat it like a journey, not just as an adventure. Treat it like a curiosity and wondrous thing. Would you attack a tiger or snake in the wild just because it spooked you? What about a human of a different skin color who accidentally startled you?
What follows is my own dissection and elaboration of how to make Mythos monsters come alive for your players, like real animals and characters should. These are taken mostly raw, with some elaboration, from my own Facebook musings on the topic. I hope they help you in your future endeavors as a Keeper. May the dice fall in your favor, and may your players ever tremble at your Lovecraftian depictions!
The biggest thing, therefore, to remember about any Mythos monster, especially the non-race and god entities like Hounds of the Tindalos for example, is that these are not just mindless animals. They are intelligent, thinking creatures. They don't just attack until dead, and they won't attack if they don't have good reason to - would you? Would a dog? If injured too badly, they will and often do run away in my games. They act to self-preserve, like any living being would, for that is what they are. They need to eat something, sleep somewhere, drink something, breathe something, amuse themselves with something, fuck something, and shit somewhere just like any creature does; sometimes their motives are as simple as basic creature comforts and needs even if those needs are not 100% the way humans would see them. A scared mother will protect her offspring, and so too would Shub-Niggurath be pissed as all hell at you if you murder her Dark Young. A curious Fire Vampire might touch and burn things first, and ask questions later, never considering that there are some creatures that surprisingly do not like to be on fire. If they are in pain, they exhibit the desire to get away from pain; if they are scared, they will go into fight, flight, or flee mode. If commanded by another being, they will do what that being says. Bottom line, they have goals and motives - often, ones that make no sense to humans, or are completely mundane.
Maybe that byakhee is trying to lick its wounds after someone shot it, or stopping at a lake so it can have some water before its interstellar flight, and doesn't know or care that the Investigators are there until it's disturbed. Maybe the shantak has a fear of fire due to being caught in a fight with Cthugha's spawn, so fire will frighten it off - or maybe it mistakes the Investigators' flashlight beams for fire and flees or attacks. The shoggoth could just think that chasing the Investigators down the icy tunnel in the frozen antarctic is a fun game of tag, and "Tekeli-li!" is an imitation of it saying "Run away!" in a joking manner simply because its former masters did that while playing tag. A Night Gaunt is trapped in a snare set by someone or something, and your Investigators happen across it, and it recognizes that they can help it. A scared and trapped Hound of the Tindalos who is starving and just wants to go home because the scary curves are scary.
Going on to Mythos races, maybe the Deep One Hybrid is part of the Esoteric Order of Dagon... or maybe he is a victim of his genetics, and is trying to warn the Investigators about the REALLY bad Deep One Hybrids. Maybe the Mi-go is actually not trying to take your brain, but noticed you really badly need surgical assistance and no human can do it without killing you - say, you will bleed out if it doesn't use its surgical precision to stitch your blood vessels neatly back together, or you've been poisoned by a Mythos toxin and it has the very icky and very disturbing antidote. In this sense, intelligent Mythos races can also be victims. Who's to say a Yithian can't go mad if it sees Azathoth's court? Knowing about it and seeing it are two very different things. What about a Deep One colony being victimized by pollution, so they try to wake Cthulhu up early, or Mi-go colonies and the impact of fracking? And that's nothing to say of the frozen Elder Things left in Antarctica and the impact of global warming!
Moving on to gods, let's use a favorite god of mine, the Crawling Chaos himself. The Big N is an excellent example of this dichotomy of how, in the Mythos, dark is not necessarily evil and blue and orange morality is the order of the day. Nyar on the whole is a Trickster God. He will act like a troll. He will behave in confounding ways to amuse himself. He is an agent of change, an agent of how random and chaotic the universe is. Does it make any sense, in any context, to then introduce Nyarlathotep as a simple villain that is out to destroy everything? Absolutely not! Sure, Nyarlathotep will try to trick your Investigators, but he also might just do nothing, or be helpful, or trick their opponents instead. In fact, he could secretly be in your Investigators' corner, right now, because whatever they're doing is interesting or genuinely useful to the universe in some insignificant way - yes, I have done that before in a game, and no, it doesn't make him any less frightening to deal with whatsoever. Sure, he's going to cause the end of humanity as we know it - but he isn't doing it because he gives a shit about bringing harm to the mere mortals. He's doing it because Azathoth Wills It, or because he's bored, or because he thinks it's fun to watch humans hang themselves with their own rope. And he's for damn sure doing it to other Mythos races, too - there is, somewhere out there, a gang of maddened Cats of Ulthar that worship the God of a Thousand Masks.
That said, who is to say what he stands for is evil? It's not - it's just change on the whole. In that case, then, anything that brings about change is Nyarlathotep's modus operandi - effect change, and see what happens. In fact, that is exactly what my Nyar does: he poses dilemmas, messes things up or changes some aspect, and sees what the humans do about it. He also tempts, sets traps, and toys with people specifically to goad them into some action favorable to him for some reason (which again, could be as simple as "You're in my house, please leave" or "I'm bored and this is fun".) He's not actively bringing ruin to the earth - humans will do that themselves - he's just giving them the rope to do it. What the humans do with that rope depends - do they use it to lynch other humans, or do they use it to lift other humans up? Do they use it to destroy, or to build? Chaos is an inherently neutral force - like life, death, and time. In my current run of MoN, for example, he wants his cults to be dealt with by the five puny mortals - the cults are self-serving and can't even get along without bloodshed, and Nyar is sick of it. Thus, just like he did with the Carlyle Expedition to set up the Great Gate, he used Jackson Elias to bring the right people into the right place at the right time to teach his cults a lesson in humility, and he will continue to do so. Not only are the cults his pawns... the Investigators are, too. He's got a long game going, and he is playing both sides - mostly out of, you guessed it, a desire to amuse himself. No matter who wins overall, he succeeds. Gods often work in mysterious ways indeed...
In more layman's terms, Evil is not a thing the Mythos deals with, because it's beyond human Good and Evil. In the case of gods and Great Old Ones, I posit that these are therefore only malevolent in the same way that a meteor wiping out the planet or a hurricane is malevolent. They are things humans cannot really comprehend or predict that well, but are terrified of, and can observe. Things that by their definition cause untold destruction and human misery, without ever really intending to. Does a hurricane care how much damage it causes? Does gravity have any inherent special meaning when it pulls an object you dropped to the ground? Absolutely not.
My point is, monsters in the Mythos have motives, they are intelligent and can reason. Those motives are not necessarily harmful, but to a human they can look very malevolent. No matter if the Deep One colony attacks your ship, or saves you from the shipwreck, seeing a fish-like amphibious man from the big, spooky ocean is still scary - and good luck trying to communicate with them, understand their methods and ways of doing things, etc.