Is figured this title would get your attention.
Now I like this class and don't see any problems with this in an occult campaign, but I don't see how it thematically fits either. What makes the kineticist occult-themed? Encouragement of non-standard problem-solving relative to the genre?
Then again, Cthulhu was killed by ramming a boat into him...
I think it has to do with flavor. We've all latched onto the Avatar/Korra comparisons, and with reason, but I think we were also meant to think Carrie or Firestarter or even some versions of Jean Grey. Those are thematically very different from benders, but the mechanics are identical.
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Well, Cthulhu is CR 30, it would be difficult (but not impossible) for a single PC using pathfinder rules to defeat him.
Korra on the other hand displays access to the Kineticist capstone, due to being the avatar and being able to manipulate all four elements, thus we know that she is at least Kineticist 20.
The biggest issue for defeating Cthulhu would be his immortality, which requires you to beat Cthulhu at least twice, the second time being within 2d6 rounds, which assuming an average of 7 rounds, Korra would have to do 117 damage per round. Additionally, Cthulhu is immune to Korra's main element, cold and resists each of the other elements by 30 points in addition to having 41 SR. Metal Blast will likely be her best weapon against Cthulhu due to his resistances, she would do the most damage with this blast if she utilizes Kinetic Blade and Improved Vital Strike and possibly Mythic Vital Strike, resulting in 60d6+60+3x her con mod in damage a strike, assuming 50 Con this would be 60d6+120 or 330 damage per round before DR, which goes down to 310 after Cthulhu's DR kicks in.
At 310 DPR, Korra would take approx 3 rounds to defeat Cthulhu the first time, which means as long as Cthulhu-GM doesn't roll low on the 2d6 rounds of being staggered after Cthulhu resurrects she could theoretically solo Cthulhu. However Cthulhu isn't going to simply stand there and let Korra beat on it. If Korra takes all of Cthulhu's physical attacks each round, she is looking at 8d6+46 and 8d10+136, meaning Korra takes 254 damage per round.
Going back to our 50 con assumption, Korra has approx 545 HP (assuming Kineticist 20/Mythic Champion 10) before toughness, so while Cthulhu is not staggered it will 3 round Korra, which is the same as the 3 rounds it will take for Korra to down Cthulhu. Additionally, Korra needs to make a DC 40 Will save at the beginning of the combat or die, and Cthulhu isn't likely to simply full attack her every round, he will likely use his special abilities.
So back to your question as to how Korra would deal with Cthulhu, I would have to say luck.
As for the question actually posed, while I don't feel that it fits the idea of 'occult' I don't really give a damn either. There's not going to be some -better- spot to put in a bending class.
As for Cthulhu, and silly theorycrafting, I'd say Korra isn't necessarily level 20 (lord knows she gets her ass kicked all the time), but more that there is an 'Avatar' archetype that gives you Omnikinesis in exchange for the inability to be resurrected. But her soloing Cthulhu is moot, as the Avatar is never really at their best unless they've got their adventuring party around them.
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As for Cthulhu, and silly theorycrafting, I'd say Korra isn't necessarily level 20 (lord knows she gets her ass kicked all the time), but more that there is an 'Avatar' archetype that gives you Omnikinesis in exchange for the inability to be resurrected. But her soloing Cthulhu is moot, as the Avatar is never really at their best unless they've got their adventuring party around them.
She doesn't exactly play by the rules of this class either, she would have a bit of Monk/Brawler as well as Summoner (Synthesist) to incorporate the martial arts and avatar state.
Plus let's face it, she'd lose a few times, to Cthulhu's minions, get rescued by team avatar, team avatar kicks butt, finds out Cthulhu is the man in charge and confronts the horror beast, fight progresses, a couple of episodes later Korra goes into the avatar state and somehow miraculously wins but then she is the main character, that is what she is supposed to do.
She has impressive abilities, but she doesn't have to be some CR 30 monstrosity that can beat every single monster in pathfinder because of this. Most of her abilities can be represented by using some levels from classes and mythic tiers.