The Assyrian Empire first began to grow due to an increase of wealth and then due to an increase of military strength.
Assyrians established the trade colony of Karum Kanesh in the Port of Kanesh, which became the most profitable trade center in all of the ancient Near East. Merchants who resided in Ashur would trek from Ashur to Kanesh, set up their trade business, complete with employees, then travel back to Ashur where they would continue to reside and supervise.
As wealth grew in Ashur, the city expanded, eventually leading to the rise of not just the city but the entire empire. As they traded with Antolia, they also gained more access to iron with which Assyrians could fashion weapons. The development of iron weapons soon led to military buildup; military buildup led to the ability to drive competing civilizations out of Ashur and to lay siege to neighboring regions.
First, Shamashi Adad I, king of Assyria from 1813 - 1791 B.C., drove the Hatti and the Amorites out of Ashur, making Ashur the capital of Assyria. Later, King Adasi (1726 - 1691 B.C.) was able to conquer the Babylonian Empire, which held claim over Ashur, regaining total control over Ashur once again. Soon, King Ashur-Uballit I (1353 - 1318 B.C.) was able to conquer the Mitannia and Hittitea, who held power over Mesopotamia and thus over Assyria as well. Later, King Adad Nirari (1307 - 1275 B.C.) was able to conquer the rest of Hittite land and even deported the remaining Hittites. Throughout generations, military warfare continued to expand the empire to cover all of Babylon, extend into Canaan, and across to the coast of the Mediterranean.