Aemond says he is next in line to inherit the Iron Throne after his older brother, Aegon. According to the Westerosi laws of inheritance, Aemond would rank behind Aegon's children, or at least his two sons.
Aegon is crowned as king of the "Andals, Rhoynar, and the First Men." The Rhoynar are the ancestors of the Dornish. At this point in Westerosi history, Dorne is still an independent principality, and not under the rule of the Iron Throne.