The first official Tlatoani (that is, ruler) of the Mexica people (the dominant power in what we call the Aztec Empire or the Empire of the Triple Alliance) was Acamapichtli (ruled 1372-1391). During his rule and that of his son Hitzilihuitl (1391-1415) and grandson Chimalpopoca (1415-1426), the Tlatoani presided over the growth of the city of Tenochtitlan from a small town to an important city in the Basin of Mexico region. During the reigns of these Tlatoani, the Mexica were vassals to the Tepanecs, the greatest power in the central Basin, ruling from the city of Azcapotzalco. During the reign of Chimalpopoca, the Tepanecs granted rulership of the Acolhua city of Texcoco to the Mexica, as a prize for their help in defeating the Acolhua city-states in battle.

The Empire of the Triple Alliance was "founded" in a sense, by the next ruler, Itzcoatl (1426-1440) who was the youngest son of Acamapichtli and therefore uncle to Chimalpopoca. When the powerful Tepanec ruler Tezozomoc died in 1426, the Mexica supported his son Tayauh, but his brother Maxtla was victorious in the struggle to succeed their father. Around the same time, Chimalpopoca was killed (possibly his death was related to the Mexica backing the loser in the Tepanec succession struggle). Itzcoatl, an experienced general of the Mexica, was chosen as the new Tlatoani. Behind the scenes, two other powerful leaders contributed to the creation of he empire, Itzcoatl's nephew Motecuhzoma Ilhuicamina (who would later succeed his uncle as Tlatoani) and Tlacaelel, the Cihuacoatl (literally "woman serpent") which position was something like a regent (when the Tlatoani was away fighting or conducting diplomacy) and Minister of Internal Affairs. Itzcoatl formed an alliance with the ruler of the Mexica's erstwhile Acolhua vassal, Texcoco (a Tlatoani named Nezahualcoyotl), who had been exiled by Maxtla to what is now Puebla, where he sought allies among the city-states of Puebla and Tlaxcala. Maxtla also blockaded Tenochtitlan and demanded tribute from the Mexica. Itzcoatl found more allies in Tlacopan, a Tepanec city-state that was rebellious against Azcapotzalco. Nezahualcoyotl found support in the city-state of Huexontzinco. Together, this alliance fought and defeated Maxtla and his Tepanecs and became the dominant power in the Basin of Mexico. Tenochtitla, Texcoco, and Tlacopan became the three "heads" of the Triple Alliance, with the Mexica of Tenochtitlan as the clear senior partner. During Itzcoatl's reign, the Alliance began the creation of its empire by conquering the Basin of Mexico and the neighboring areas now known as Morelos.

"Tenoch" is said to be the predecessor of Acamapichtli and the founder of the city of Tenochtitlan around 1325. However, with the rise of the empire, the Mexica elite, presided over by Itzcoatl, Motecuhzoma Ilhuicamina, and Tlacaelel, made a conscious effort to create a new official history that glorified their ancestors and their origin and migration myths. Tenoch was a part of this new history, and it is uncertain to what extent the figure is based on a real person, and to what extent he is a creation of state mythology.

Source on Aztec history: "Smith, Michael E. 1996. The Aztecs. Blackwell, Cambridge, Massachusetts.

I don't really have a source on this interpretation of Tenoch to hand, but Charles Mann in 1491 does write about the creation of the official history of the Mexica state during the reign of Itzcoatl.

EDIT: Clarity and spelling