This is Gustaf right? It doesn't really seem like he's wearing green....It looks more like his red robe covered a beige skull undergarment?
The only thing we know for certain (the "facts" of the case) are that he was the first king of Hyrule, he had a good relationship with the Wind Tribe, his spirit appears to help Link on his quest to find the Wind Element which he knows is hidden with the Wind Tribe, he helped create the Kinstone Door that blocks the entrance to the Veiled Falls (and the Wind Tribe that lives above it), and he is buried in the biggest tomb in the Royal Crypt, meaning his legacy was well-respected.
If he is indeed wearing green, this may indicate his use of Wind Magic. Wind magic in the Zelda series is often associated with the color green (as in the Wind Element). Maybe the king helped cast whatever spell that allowed the Wind Tribe to go to the sky. At the very least, he helped create the door that blocks off the Veiled Falls and thus keeps the Wind Tribe safe from danger (presumably the danger was the Masked Monsters that were bound in the Bound Chest during the War of the Bound Chest.) There is also something to be said for the fact that the Wind Tribe's old home (the Fortress of Winds) is shaped like the skull that it looks like King Gustaf has on his clothes, and the second floor is shaped like a crown.
My theory is that King Gustaf, the first king of Hyrule, was a sixth-generation descendant of Zelda from Skyward Sword who lived during the War of the Bound Chest in which the Masked Monsters invaded Hyrule and were ultimately defeated by the gifts granted by the Minish. King Gustaf, during his rule, formed a good relationship with the Wind Tribe, wind mages of the Gerudo who guarded the Wind Element at the Fortress of Winds, built with the help of King Gustaf. Upon the invasion of the Masked Monsters, the Wind Tribe were forced to flee to the clouds above the Veiled Falls with the help of King Gustaf, who may or may not have learned wind magic from the tribe, and it was ultimately he who created the Kinstone Door to protect the Wind Tribe from the invading monsters. Then the Hero of Men appeared, the Picori were sent by the Gods to deliver the Picori Blade and the Light Force awoke within Zelda who used it as a sealing power. Then Gustaf and Link became good friends, and that became a generational relationship: the King and the descendant of the Hero of Man (which is why King Daltus and Smith are good friends).
The more outlandish theory involves a subgroup of the Wind Tribe breaking off and becoming the Interlopers, and, instead of Gustaf participating in the War of the Bound Chest, he in fact was king during the Interloper War. Not much changes about that, except instead of the Gods sending the Picori, they sent the four light spirits and instead of the Light Force awakening within Zelda, the Triforce was sealed by Gustaf's heir, Rauru. That one is a little less grounded.