The Aztecs have a large force of infantry, with most being melee units. The War Hut units are cheap and light, but still somewhat effective. War Hut units are easily massed, and should make up the majority of an Aztec Army, dealing damage and taking the heat off of stronger units. The Nobles' Hut trains elite forces which are mainly slow, but hard-hitting. Also their army is population effective because of all their army is infantry, each unit only takes up one or two (and rarely three) population slots.The Aztecs have two very powerful infantry: The Jaguar Prowl Knight, which is not only good against both cavalry and heavy infantry but can also use stealth, and the Skull Knight, which does great damage and has an AoE attack.
Despite the differences, Aztecs play quite similarly to the other armies, with the Arrow Knight being an Aztec anti-artillery unit, Coyote Runners taking the place of melee cavalry, and the Eagle Runner Knight replacing ranged cavalry. The Warrior Priest can not only heal troops, but it also fights well, and it counts as two units when dancing at the Fire Pit.
The Aztecs have the best navy of the Native Americans, with the Tlaloc Canoe dominating the seas. As a Canoe, it also counts as a Fishing Ship as does the rest of the Aztec navy, and so can gather Food or Coin during peacetime. It's not as powerful as the European Frigates or the Asian Tekkousen or Fuchuan, however, but are much cheaper. With the Water Dance they are capable of taking on European ships.
A large advantage of the Aztecs is the wide variety of cards in their deck. Their sheer versatility is what makes them a danger as they can boom, rush, or turtle very efficiently.
A very effective steamroll strategy is to create an army until you have almost reached the population limit. Then, after at least 30 minutes have passed, you research all the "large button" technologies which give you 15 of almost every type of soldier, plus 7 skull knights. Thus you have a gigantic number of soldiers at your disposal, which can be used to steamroll an enemy with very little tactical effort.
A good strategy is to fight in numbers since their entire force is infantry. In the early game, your army should be composed of Puma Spearman for their cheap cost and effectivness against cavalry with Macehualtin in large numbers to be powerful against infantry and some Coyote Runners to use against enemy villagers to hurt their economy. In the mid game, your early game strategy can still be used but now you can include the knights from the nobles hut to help. Now Arrow Knight and Eagle Runner Knight should be utilized to improve your ranged fire power against buildings, artillery and cavalry of opposing armies. The late game strategy that is powerful is a front line of skull knights with your villagers and warrior priests all at the fire pit, followed by a force of coyote runners to close the gap on artillery with ranged support of arrow knights preferably for their range and further anti artillery power which the skull knight front line is vulnerable to. Also if enough coin is available, send Zapotec and Mayan allies from the Home City to make the army go beyond its limits.
The lack of artillery renders Aztec firepower less effective against massed units and building structures; while the Arrow Knight is the strongest anti-building/artillery unit, Aztec armies possess nothing comparable to Falconets or a Great Cannon. The Skull Knight possesses a melee area attack, but can only be obtained through Fire Pit dances and Home City shipments - both activities consume time and effort.
They have less ranged capabilities than most armies, and will do most of their fighting in melee - and most of their units are slow enough that enemy artillery will deal considerable damage.
Also note that even though they have the strongest Native American navy, they are still at a disadvantage when facing European or Asian vessels - they need a large force of Canoes to counter a few Frigates, but the disadvantage can be largely, or entirely, offset by cards.
Their biggest weakness is their dependency on cards so a blockade will damage their ability to fight effectively.