The best strategies for defeating bosses in the Souls games typically involves staying as close as you possibly can to the boss, circling around to their backside, stabbing them in the butt, and rolling under their attacks when necessary. A few bosses in the series will break this convention and force the player to have to find different ways of defeating them. Dark Souls III has one particular boss that makes this go-to strategy almost completely futile. If you're able to circle-strafe around the feet of the King of the Storm boss in Archdragon Peak and then still beat it, then I am impressed.

The Nameless King and his storm drake require a different strategy than the typical Dark Souls boss.

Why you should keep your distance

The King of the Storm is one of the rare bosses in the series that actually requires the player to keep your distance from the boss and avoid getting in too close. There are actually quite a few reasons for this, but there are two that stand out as making this boss particularly frustrating for any player who tethers themselves to the boss's feet. The first is that you'll end up forcing yourself into battle with the most ever-present of Souls games' most formidable foes: the camera. Locking onto the enemy is impossible when directly below him, and so you won't be able to see any of what he is doing. He also moves very quickly and can cover a lot of space, so you'll end up spending a lot of time having to micro-manage the camera as you try to manually keep it focused on the dragon.

The second reason is that the dragon has a potent defense against this sort of strategy: its fire breath. If you hang around under the dragon's belly, it won't wait too long before flying up into the air and unleashing an area of effect fire breath attack that can often mean certain death if you get caught in it. Even a health bar that stretches most of the way across the screen can be more than halved simply by being hit with this attack. But that isn't the worst of it. The attack will also knock the player prone, but the fire spray also persists long enough that it can frequently trigger a second hit once you stand up. This second hit will almost certainly kill you.

Getting caught in this flying fire breath attack means almost certain death.

These two reasons should be enough to keep you out from under the dragon, but there are other reasons why keeping your distance is the better option....

UPDATE SEPTEMBER 1, 2016:
The release of Dark Souls III has finally answered this question. That game has rendered the speculation below completely moot and invalid. However, this post will remain preserved for posterity.

There is considerable debate within the Dark Souls fan base regarding whether or not Knight Solaire of Astora may be the forgotten son of Lord Gwyn, and the God of War. The game contains many references to a forgotten God of War, who was the son of Lord Gwyn, the God of Sunlight. The primary source for this is the item description for the Ring of the Sun's Firstborn:

"Lord Gwyn's firstborn, who inherited the
sunlight, once wore this ancient ring.
Boosts the strength of miracles.
 
Lord Gwyn's firstborn was a god of war,
but his foolishness led to a loss of the
annals, and rescinding of his deific status.
Today, even his name is not known."

The game doesn't specify what the God of War did to be stripped from the annals, but the blunder cost him dearly. As a punishment, Gwyn and the other gods rescinded the God of War's diefic status and expelled him from Anor Londo.

But it didn't end there. Based on the content of the game itself, it appears that the gods also removed all references to him. This included removing or destroying any statues depicting him and redacting his name from records. Both his name and face are lost to history, as is his fate. Was he cast out of Anor Londo? Was he made mortal? Was he cast out of the world, entirely?

The gods made an effort to eradicate all records and traces of the God of War.
Statues of him were removed or destroyed all throughout Anor Londo and Lordran.

If the Warriors of Sunlight covenant (lead by the God of War) existed prior to the God of War's expulsion from Anor Londo, then it would also stand to reason that the gods disbanded the covenant. It's highly unlikely that gods would have permitted the covenant to continue to function, since its followers would be able to continue to teach of the existence of the God of War. These followers would likely have been forced to renounce the covenant or become an underground cult. The fact that the covenant still exists and has followers is likely due to the waning influence of the gods after Gwyn left to link the flames and the gods were forced to abandon Anor Londo.

The Sunlight Medal does offer one clue as to the fate of the God of War and his covenant. It suggests that the forgotten god still lives, and still watches over his followers:

"This faintly warm medal engraved with the
symbol of the Sun, is the ultimate honor,
awarded to those who summon the Warrior
of Sunlight and complete a goal.
 
The symbol represents Lord Gwyn's firstborn,
who lost his deity status and was expunged
from the annals. But the old God of War
still watches closely over his warriors."

In any case, we meet one NPC Warrior of Sunlight during the game: Solaire of Astora.

Solaire doesn't reveal much about himself. He is a Warrior of the Sunlight who has a strong reverence for the sun itself. In fact, he is on a possibly futile attempt to obtain his own "sun"...