I think the Aztecs are fine. I just finished a game in which I was Aztec (Epic, Monarch, Standard, and having NEVER played Aztecs before.. randomly got assigned them) and EASILY won, probably the easiest victory I've had in a while. What I did was find that I had no copper, so I hastily chopped myself a Great Wall (I was next to marble and within reach of stone early on). I founded 2 cities and started cottage spamming on riverside floodplains even before I was going to go on offense. Once I had a jaguar out, I DoW on France near me, pillaged their metal, and parked it on a forested hilltop. After barracks building and pumping out more jags, I captured a French expansion city and parked some more Jags near the other French cities and pillaged their horses and other stuff to make money. I was going to keep pressing France, but I didn't have the firepower to knock out any other French cities without cats, and my cities were busy building courthouses and infrastructure instead of reinforcements. I also needed French techs via extortion in order to catch up on the tech race.
Now say what you will about the Aztec UU sucking because it is basically a weaker Swordsman that doesn't require metal, but thank goodness for that, because I didn't get iron until so late in the game that I had war elephants by that time. (Normally I have a fast axe rush and capture an enemy capital right away, but without copper I couldn't do that. I researched IW as soon as I could and didn't bother expanding towards iron because I just needed the tech in order to make jags.)
While I'm beelining for cats, I DoW my other neighbor, the Mongols, since they were getting way too big, and I want to park some units on their territory to slow them down. It works, and we have a sort of cold war where I can't attack their cities but they can't attack my pillagers either. Then I got Construction and made some cats. But wouldn't you know that they JUST got Feud right as I got there. So I could only capture 2 Mongol cities before being forced to declare peace because their freaking longbowmen were slowing me down too much, and also because I wanted to pick on France which didn't yet have longbows. You've got to strike while the iron is hot--and while the other civ doesn't have longbows!
So I re-DoW on France and am doing a very good job of finishing France when Khan demands tribute. WTH, I took out 2 of your cities not so long ago and now you demand tribute? Anyway, I say no, gambling that I will take out France by the time Khan attacks my other flank, but that's not quite how it worked out.
Khan declares war on me before I knock out France, and he managed to take back his capital city with a wave of keshiks. I stubbornly try to fight a 2-front war, but it's clear that I just don't have the manpower to take out the remaining French cities while warding off Khan. Worse, from completely out of the blue, the map leader Egypt DoW on me.. probably Khan had enough money and tech to bribe him, since Egypt is rarely aggressive. I will NOT be dogpiled. Ever. That's how you lose games.
So I IMMEDIATELY bribed Spain, Germany, AND Rome to attack Egypt, all using the same tech to bribe them with (Philosophy). (This is why I cottage spam early--get a tiny tech advantage to save up for a rainy-day bribefest.) I fought only a few stacks of Egyptian soldiers before there was a total silence from them--I could only imagine the carnage that my allies were inflicting on Egypt's home turf. Actually I didn't have to imagine it; I just couldn't SEE it since they were so far away. But the plunging yellow line on the Power graph told the story.
Fortunately by this time Bismarck suddenly DoW on France and took out all of the remaining French cities while I was regrouping. Khan put up a huge fight, and I had to sue for peace after I re-took the Mongol capital (that stupid city traded hands a lot of times) since my anti-French army took FOREVER getting back home and to the other front, and I knew that I couldn't hold the Mongol capital against the big Mongol keshik stack that just popped into view. But I redeclared on Khan once I got grenadiers and trebuchets; he folded like a house of cards in the face of a hurricane. Most of my units by this point were war elephants to counter the keshik-spamming Mongols, by the way.
At this point the game was mine: Nobody else had the real estate I had, and I was tech co-leader with Spain. Germany did get a few French cities, and Rome and Spain got about 1/2 of Egypt each, with Germany coming late to the party due to distance but managing to capture 2 Egyptian cities. But I had my own area, most of France's area, and most of Mongolia's area, making my empire triple the size of, say, England. Churchill never invaded anyone and therefore didn't grow. The only nations left were Spain, Germany, Rome, England, and me, and everybody was Friendly towards me except Churchill who was merely Pleased since he was friends with France, whom I warred with so much. Those +diplos for "mutual military struggle" and some "helped us/gave us tribute" points added up, and I spent the entire game without a religion to appease Spain.
Later on Rome invaded England, and everybody dogpiled onto them except me, until someone asked that I join in, and I said yes, and 2 turns later England was completely conquered anyway so it just gave me even more diplo points for no cost.
At this point I turned into a builder for a while to recover the tech lead and to see who would emerge as the top dog among the AIs. It turned out to be Spain, surprisingly, so I bribed Rome and Germany to beat her up. I tried to send some outdated units to Spain to gift (for Izzy to upgrade cheaply and thus to prolong the war), but Spain made peace with everyone before I could get there in time. Then I got Rome to fight Germany. I briefly lost my entire tech lead by bribing everyone to fight each other (3 techs each to Rome and Germany, but they were the same techs), but that was the last big bribefest I needed, as by that point I was mostly done building infrastructure, had rebuilt my tech lead to something ridiculous, and was starting to crank out a lot of units. (After conquering Mongolia, I only built a small trickle of units here and there to keep my power rating up, not that I needed to bother since I was best buds with everyone anyway.) Furthermore, I could have gotten an easy U.N. victory because everybody hated each other except me; everybody loved me and would have voted for me. But I don't like U.N./diplomatic victories.
The key to this game like many others was my early crippling of my neighbors while keeping more-distant civs happy. You should be HAPPY when those father-away civs demand tribute from you--unless you are about to attack them anyway. Because after a while those tribute modifiers turn other civs into Pleased and that means that they're less likely to DoW on you. My favorite part of that Aztec game was when Julius asked for 190 gold, I gave it to him, which instantly turned him Pleased, then I immediately asked if he could help me as his friend and he said yes and gave me 370 gold back.
By the way, dogpiling is something you almost always want to avoid. I cannot stress this enough. Egypt is the poster child for this: Egypt beat me to the race for Liberalism, but Egypt did not have the friends that I had. You need either a LOT of real estate or a GIANT tech lead or both if you want to take on multiple civs. Egypt had the biggest empire and a substantial tech lead on everyone, but it buckled pretty quickly after I killed its first invading stacks and got 3 other civs to attack Egypt while I took on Khan. Also, remember you can coordinate attacks. I remember asking Rome and Spain to both attack Pi-Ramesses (sp?) in order to coordinate their efforts, and a few turns later it was captured. I stopped getting people to coordinate their efforts after a while because it was in my best interest to keep Egypt in the game--I didn't want it totally dead, just small and lagging and not a threat. Unfortunately my allies wouldn't quit attacking and got big off of Egypt. But by now you know how I took care of that.
P.S. Open your borders ASAP. You need those diplo modifiers right away. I don't even bother trying to block people off most of the time unless it's easily done. In my current game I did block off my neighbor on one flank because it was fairly easy to do, but I gave him open borders the moment that I didn't need it anymore. But in the Aztec game: what was I going to do, spam three cities to block Khan off while warring with France? Not if I wanted to keep my tech slider at 70% research! Khan got big because I didn't expand at all in his direction, but I didn't care because he was my next victim anyway. So open your borders unless it really does make sense otherwise. And take lopsided tech trades if your trading partner is probably going to research or trade for it soon... the more lopside the trade the more likely you get +diplo points. Having your neighbor DoW on someone other than you is priceless early on.