According to the Fire Emblem Complete, Andorey's mother is different from Brigid and Edain, this indicates Ring either had two different wives or kept a mistress.
During a pilgrimage to Blaggi Tower with his family, Lord Ring's ship was caught in a fierce storm. Ring managed to obtain his wife, Edain, and Andorey, but the energetic Brigid ran off, and was unable to be found before they evacuated the ship. Despite future efforts, Brigid was not found by Lord Ring. 
During the Isaachian-Grannvaleian War, Lords Ring and Byron accompanied Prince Kurth to the front lines, with Ring not bringing the Yewfelle along with him, and bringing Andorey with him. Upon returning from the war, Prince Kurth is assassinated in a scheme that is devised by Dukes Reptor and Lombard, of Freege and Dozel respectively. The blame is then placed upon Duke Byron, with Ring being accused of being one of the conspirators.
Not long after this horrific revelation, Ring is murdered by his own son, Andorey, who does so in order to take control of the Dukedom of Jungby. In Chapter 5, Brigid and Edain avenge their father's death by slaying the treacherous Andorey.
It is also shown that, depsite Lombard being one of the schemers against Sigurd, he does think of Ring in a higher light, as he is shown to be off-put by Andorey's actions, and says for him to rest in peace.
Little is seen of Ring's personality as he is never encountered ingame, but he would appears to be a beloved father to his daughters. According to a villager in the prologue, Ring's relationship with Andorey was more tumultuous, at least by that point.
Ring's personality is slightly expanded in the Oosawa manga adaptation. While he appears to be rather level-headed when dealing with politics, his interactions with Byron bring out a more temperate and moody side to him. Ring is also very protective of his daughter Edain, as can be seen when almost panics when he mistakenly thinks that she has eloped with Sigurd.