King Arthur arrived to Gielinor in 132 of the Fifth Age, where the Knights of the Round Table were to remain until their aid was needed again in Britain. (When speaking to the Guard at the beginning of the King's Ransom quest, when you ask if Arthur and his Knights's disappearance could be due to their returning to their own realm, the Guard will enigmatically respond that there is no evidence to suggest that.) At the time King Ulthas still held the throne of Kandarin. Once in Gielinor, King Arthur sought to find a new castle to house his Knights.
King Ulthas, wishing to aid King Arthur, spoke with Lord Sinclair, an important political figure who had taken up residence in a mansion east of Seers' Village. Sinclair agreed, deciding Arthur's need was greater than his. He left the mansion to Arthur, relocating himself and his children to a small mansion near the southern border of the Fremennik Province. His children, though, never forgave him for leaving their home, eventually leading to his murder.
King Ulthas died shortly after, leaving the throne to his sons Lathas and Tyras. King Lathas, the ruler of East Ardougne, disagreed with his father's decision to give the Knights Camelot, and to this day wishes to reclaim the mansion and have the Knights killed. To do this, he allied himself with the mourners of the elvenIorwerth Clan, whom he believes will aid him in his future attack. He along with the mourners was defeated by the adventurer as part of the final defeat of the Iorwerth clan and the restoration of Prifddinas, the Elf City.
In medieval British lore, King Arthur was a legendary ruler of the Celtic Britons who supposedly defended the British Isles against an Anglo-Saxon invasion.
King Arthur can be given a poison chalice from Stankers, the coal truck manager, to complete an easy Seers' Village Task. Once given, he quotes the Grail Knight from Indiana Jones and The Last Crusade.
His chathead shows him with a short brown beard but his NPC shows him with a long red beard. His chathead also shows a thin, detailed crown, however, his ingame avatar has a very thick and undetailed crown.
The world that Arthur arrived from is presumably our own, as the characters in Camelot are sourced 'as is' from the Matter of Britain and Arthurian Legends. Arthurian legend also states that King Arthur was not truly killed by Mordred, but carried off to another world, biding his time and ready to return. The 'other world' in this case would be Gielinor.
^Fortress guard, "Black Knights' Fortress", RuneScape. "I have no idea, but it's a great victory for Lord Daquarius. To take such a mighty supporter of the White Knights down... Truly brilliant."