Hi,
I've tried last night my first game on deity. I choose Aztecs, on Continents, standard size and turn, 8 players, Australia patch. Settled in place, meet Scythia and Stockholm (west), Arabia (north), Palenque and Buenos Aires in east. Get envoys with both Palenque and Buenos Aires. Build slinger, slinger, builder, slinger and beeline archery, build two pastures and a quarry, destroy one barbarian camp between me and Palenque and I managed to build 4 archers and another jaguar by turn 35, third jaguar is on the way, got a trader with Palenque. Once Scythia declared war and attacked Stockholm, I moved my army toward them and dow on Scythia, I thought it will be easy with a part of their army attacking other city. But things get worse - Tomirys had already an army twice as mine - including one horse archer and two horsemen - she used them to flanked my archers , and I got mauled pretty bad - I lost two archers and one jaguar. From the snowy south came a barbarian scout that detected me, so I've renounced because I can't hold up the position. What went wrong with my strategy? Should I delayed my dow and try to settle the second city between me and Tomirys (one with two luxuries that might help me for a further war)? Even with two cities, I don't know if I can produce more troops than Scythia's 4 or 5 cities.
I'm getting more convinced that a super early DoW is really strong. When I first meet a civ, I always try to exchange city info, then send every unit I have to scout out their area. Then I steal the most I can get away with, be it a settler (hopefully), or a builder or two. So I almost always DoW by about turn 10.
With this strategy, I'm liking going for craftsmanship first, just hard teching it, not even waiting for the boost. Agoge is just so strong for early rush. If you meet a culture city-state, I feel it's almost overpowered. Doing this, you can focus completely on building your army (mostly slingers). I ignore even building builders, settler, or a trade route. You'll get most of the boosts from the cities you'll take. Barbarians are not really a problem since they won't be able to pillage or plunder anything, and they can't take your capital.
Tomirys is probably one of the hardest to do an early rush against (also Gilgamesh). Since a lot of her strength is ranged based because of horse archers, I might suggest doing more warriors instead of as many archers as you normally would. This is especially true for Monty. The Eagle Warrior is just super strong. It's another reason you can ignore building builders in your capital. Since she has a larger army, try attacking her fringe cities first, or at least where her army is not. The first city will be the hardest to take, but after that it will be easier, so it's ok to lose a couple units.
Also, put off finishing teching archery until you're pretty much done building slingers. Slingers are just so much cheaper. I usually build about 5 slingers first, then ~1-2 warriors, then maybe more archers as war goes on. For Aztecs specifically, maybe 3-4 slingers and then 3-4 Eagle Warriors.
I'm not sure there's much more you could have done, except maybe an earlier DoW. Or you could wait for her to declare on you, kill her units, then take her cities. Alternatively, you could turtle for a while and focus science. You're next best bet for a war is getting renaissance/industrial units. Those science city-states will be helpful for faster techs.
edit: fixed my incorrect usage of Jaguars instead of Eagle Warriors. That's my civ V coming out.