King's Haki is simply Haki that's stronger than Armament, or a buff to Armament. There's probably a limit to what Armament Haki can do, King's Haki allows one to have stronger Haki than they'd otherwise have with just Armament.
As for Black blades, Haki is almost definitely involved. Mihawk told Zoro he needs to ensure his Blades don't get nicked, it's a shame as a Swordsman to have a nicked blade, so Haki plays a role.
Armament Haki makes your blade stronger meaning it's less likely to get nicked, so King's Haki would definitely make the blade far stronger than with just Armament.
Finally, One Piece is full of Magic. Swords can recognise their user. So, it's not farfetched to think that a Sword can acknowledge someone as being the strongest Swordsman, therefore, in my opinion, it's about proving to the Sword through clashes with other swords, that you're the strongest, at which point, the Blade decides to become black.
There is also something that doesn't get pointed out:
- King said he's not a Swordsman because he doesn't have a Swords Martial art, he doesn't have a discipline, he just fights to win anyway possible even if it requires biting the opponent.
This is important because Zoro is a Swordsman, he has a discipline, a Sword based martial art he follows.... even without a Sword, he has a Swordless sword style. Just using a Sword doesn't make you a Swordsman, it just means you're capable of using a Sword at the very least.
Vista is even regarded as the strongest Swordsman within the WBP despite Whitebeard also being someone who used a blade, so possessing this one-mindedness of using a blade/blade martial arts is important in being recognised as a Swordsman.
This means if a Sword doesn't recognise you as a true Swordsman, then you're unlikely to be able to attain a Blackblade no matter how strong your Haki is... which would explain why the 2 strongest characters - Roger and Whitebeard never acquired one.
Pretty sure Shanks is also said not to be a true Swordsman as well, he likely fights with other weapons whereas a true swordsman will only fight based of off Swordsmartial arts like Zoro does.