Supposedly King has escaped situations a lot of times out of bluffing and pure luck. Is this luck natural or does he have some sort of "luck superpower"?
Well I wouldn't consider it to be a Superpower. If that were the case, he would be lucky in every single thing he did which isn't necessarily true. Also, you cannot entirely consider it to be luck as King himself puts on a very intimidating face under several circumstances and also, is capable of bluffing his way out of many difficult situations.
The character has been written in such a way who tends to be sort of lucky when it comes to him fooling the "Hero Association" and the media with regard to his strength.
When Saitama first confronts King in King's apartment, King's internal monologue includes a line to the effect that he just seems to attract monsters. In the anime, at least, Saitama also specifically remarks that monsters just seem to show up on King when the Giant Bird crashes the party. King, at least, considers this extraordinarily unlucky.
King is otherwise someone whose reputation precedes him to a comical degree.
The King Engine is really just his heart beating so hard and so loud from being scared that people can hear it. But people believe that it's the sound of his rage, or of his powering up, or something of that effect, so that when opponents hear it it scares them, and when bystanders hear it it excites them (because maybe they're going to see a dumb monster actually try to stand up to King and get obliterated).
Virtually anything King says or does gets interpreted through the lens of "this is an incredibly powerful and badass man".
Several examples come to mind. For one:
After spending a night gaming and being a mess from not cleaning up first and having spilled stuff on himself, King arrives at an early meeting concerning the Monster Association. The HQ staff assumes the "up all night" part means he's been battling all night, and that he is covered with the fluids of his enemies (rather than his slushy). Sweet Mask is impressed with King's dedication and stamina, recognizing him as a true and proper hero, and backs off a bit from his confrontational attitude with the HQ now that he knows someone like King is involved.
Recently in the anime you could see him playing a video game, and when called by the Hero Association to deal with the sudden uprising of monsters, tells them he can't because he's fighting something else right now, and has to go deal with the secret boss. They assume he's actually fighting an actual, real-world threat. One that surpasses what they're already trying to deal with.
(webcomic) During the Monster Association arc, King is escorting the kidnapped child best as he can out of the tunnels, when Tatsumaki rips up the entire base and brings it above ground. King panics and hides with the kid as soon as this starts. While the other S-classes complain about her reckless move and how it endangered them, Tatsumaki insults them and praises King as the only one of the S-class smart enough and skilled enough to take care of himself, the child, and deal with her move all at once, like a proper hero should.
(webcomic) Atomic Samurai requests that King show him his sword swing, as such a thing will reveal to him a great deal about the one who swings the sword. King says that he has no training with the sword, but Atomic Samurai insists it will still be a meaningful insight. An apple is placed on the ground, and King kneels with the sheathed sword in his hands, prepared to draw it. And he stays exactly that way for like 15 minutes, stricken immobile by a panic attack over how now Atomic Samurai will know he's a fake, swing or no swing. He then puts the sword down, stands up, and silently walks away without looking back. Atomic Samurai shits bricks, thinking that King's sword slice was so fast and so sublime that not only did he not see it, but that the apple "never even knew it was cut". He praises him for doing the diplomatic and honorable thing by quietly walking away, and with admiration calls King a true monster for managing such a feat, which was beyond even his imagination and skill, despite having "no training".
Even Saitama originally just assumes that King is very strong based on his reputation, and that whatever he is doing is because of that. He wonders if King bailed on the fight with that cyborg because it was boring to him; Saitama seemed kind of hopeful he'd found someone who shared his ennui from having amazing power. Saitama, however, is the only one that King has both confessed the truth to, and who has believed King about it.
(webcomic) Recently, King tried to gain real strength by seeking martial arts training. He tells Bang that he's not strong at all, and that Bang, being the wise old master that he is, surely saw through that long ago. Bang doesn't believe him at all, and thinks if he's going to train someone it shouldn't be someone strong (like King) but a normal person. Bang feels encouraged that even someone as strong as King still feels he can improve and has the humility to see weakness in himself. Every other dojo he goes to responds similarly, regardless of King's protestations of his actual weakness, thinking their style is too peaceful for King, or not nearly advanced enough for King, or that he was spoiling for a fight, etc.
While King does have a certain measure of luck (though at times it is a curse), he does seem to have developed certain skills to exploit this reputation. He knows he's intimidating to others and uses that whenever possible to get out of bad situations. He's gotten rather good at bluffing, as well, which is assisted by the fact that his reputation is so powerful that enemies will construe any (cowardly) actions he takes as the traps and feints of a supreme tactician. Including simply running away. Nothing has formally stated that King has any supernatural origins to his luck, so as-is it is just ordinary serendipity coupled with a bit of developed skill in bluffing.
As to how far I've read in the manga, I think that King is just someone who is naturally lucky. While the wiki does list his ability as a superpower, it was never really properly explained in the manga.
Characters with high luck in manga are not really unprecedented as I've found out that there are other characters from other series that are just naturally lucky. It's a trope called 'Born Lucky'. So, it's not really a surprise that King would be just naturally lucky. We can't say for sure that it's really a superpower, unless it would be explicitly stated in the manga or by ONE himself.