The dwarf from Thorin’s company who later rules the colony of Moria is Balin.
Balin is one of the thirteen Dwarves who accompany Thorin Oakenshield on the Quest of Erebor in The Hobbit. He is an older, respected member of the company and serves as an important adviser during the journey to the Lonely Mountain. After Smaug’s defeat and the Battle of the Five Armies, Balin remains associated with the restored Dwarven kingdom.
Years later, Balin leaves Erebor with a small group of Dwarves, including Óin and Ori, to reclaim the ancient underground realm of Moria, also called Khazad-dûm. The expedition establishes a colony, and Balin takes the title Lord of Moria. He is not normally called King of Moria: the settlement is a colony rather than a restored kingdom.
The colony eventually loses contact with Erebor. When the Fellowship enters Moria, it discovers Balin’s tomb and the Book of Mazarbul, which records the colony’s destruction by Orcs. Balin himself was killed by an Orc archer in Dimrill Dale.