The Assyrians are a good civilization for early rushes since their villager bonus can allow them to advance through the ages and scout the map faster than other civilizations.
The Assyrians can begin to flex their muscles as soon as they hit the Tool Age and build an Archery Range.
Their faster villagers allow to get ahead of other civilizations, both in research and resource gathering, so you can try to be the first to make it to a subsequent age, build a small army, and strike while you have an early technology advantage.
The Assyrians also have excellent archers during the early game and will likely make up a large part of your army. It's not a bad idea to get a small group of archers together immediately after you get to the Tool Age and build an archery range as the first Tool Age building.
Use them to harass and slow your enemy's resource gathering, picking off villagers one by one and retreating when melee units approach them.
With their fire rate increased, a large amount of them can wipe out an entire village within minutes. This makes the Assyrians an excellent choice in small land maps.
In major battles, use Infantry and Cavalry to engage enemy troops, so your archers can stay behind enemy lines and fire their arrows.
The speed advantage of your villagers will be blunted in a long game. Other civilizations such as the Hittite with their powerful siege weapons, the Persians with their faster War Elephants and the Babylonians with their tougher Ballista Towers will be able to field a more capable Iron Age army that can outperform your best Iron Age archery units given enough time.
The Assyrians can field an army of Legions and Cataphracts. Although they are not stellar, they can help soak up enemy fire and be used to counterattack the attacking cavalry that are effective against archers.
Heavy Catapults and Helepoli can also be fielded to counterattack other siege weapons which are strong against archers but the lack of the Engineering upgrade makes it difficult to outmaneuver enemy siege units, especially those fielded by the Sumerians and the Hittites.
Don't hedge your bets on controlling the seas either.
If the computer controls this civilization in Iron Age with sufficient amount of resources as in a standard deathmatch, the entire army will consist of Horse Archers, Priests and Helepoli. If resources, especially gold, become scarce, they will occasionally train Chariot Archers to supplement their Iron Age army. Once the computer trains Bowmen however, it is a sign that the computer already lost its military potential and will be vulnerable to conquest.