King Arthur was a Knight of The Round Table, monarch and legendary hero referenced in The Order: 1886. He was the founder of The Order, and a renowned King to England; based on the historical and mythological figure of the same name.
In Amongst Equals, Lafayette wonders if Arthur and the Lord Chancellor had fought alongside one another, in which Grayson replies, "That is the legend". This indicates Augustus is the first Sir Lancelot, now watching over The Round Table in Arthur's absence.
It's never mentioned within the game what happened to Arthur; if he's dead or alive, or if Lancelot and him left on friendly terms.
In 'Le Morte d'Arthur', Arthur waged a civil war with Lancelot upon discovering his affair with Queen Guinevere. At the conclusion of the tale, Arthur is gravely wounded after a conflict with his son, Mordred. He is laid on a barge by Sir Bedivere and taken away by women wearing blackhoods. It is after this event where Augustus presumably became the leader of The Order.
Unlike their fellow Knights, Arthur and Guinevere do not have seats at the Round Table. Instead, their coat of arms hang above their thrones overseeing the Round Table. In between their seats is a throne made of stone, and alongside Arthur's chair is a grail which holds Blackwater.