All of the kings of Tenochtitlan (the Aztec capital) and later the Aztec Triple Alliance/Empire were from the same family, and although members of the high nobility were theoretically eligible to be emperor once the position of emperor existed, no one who was not a part of the royal family ever held the position. Pre-empire, the succession of kings was strictly hereditary, but the succession laws did change one time, and that time was around 1428 when the Triple Alliance between Tenochtitlan, Tetzcoco, and Tlacopan was founded.

Starting with Itzcoatl and continuing until the death of Moteuczomah Xocoyotl, the succession was a sort of elective monarchy where a council of powerful nobles would select who they believed to be the most capable heir from a small pool of candidates including legitimate sons in the royal family and other members of the upper nobility and appoint the chosen one to be emperor. The switch owed to the fact that the last pre-imperial king, Chimalpopoca, was assassinated, and did not leave a direct heir. Itzcoatl then set a precedent by being elected based on experience and merit. (However, it may have been Itzcoatl’s idea to assassinate Chimalpopoca, so it is possible that the first election wasn’t exactly fair.) Nonetheless, Itzcoatl was able to prove his legitimacy through military success and died nonviolently in 1440. The tradition of legitimacy to rule being strongly tied to military prowess continued from there on, and was actually emphasized more as the Aztecs realized that whenever leadership changed some of their tributaries would try to break away, so a strong military leader who could keep things together was necessary. For example, the second emperor, Moteuczomah Ilhuicamina, was elected based on his experience and performance serving the role of military second in command to Itzcoatl during the war to overthrow the Tepanec Empire. For an opposite example, we can look at Emperor Tizoc, who had a short reign from 1481 to 1486, on account of his military failures which reduced the military reputation of the empire and caused several revolts. We don’t know who actually killed Tizoc, but he was killed without much complaining from the nobility and the much more successful Ahuitzotl took his place.

You can see the succession of emperors starting with Acamapichtli, a pre-imperial king, here. The numbers next to the name denote the order of kings.

Source: Aztec Warfare: Imperial Expansion and Political Control by Ross Hassig.