Going for Diplomatic Victory...
As of 500 AD, I have 5 cities and I'm massing War Elephants and Catapults for war. Slightly before this time, I bribe Shaka to attack Egypt (largely to keep either of them from attacking me). In 635 AD, my army is ready and I declare war on Egypt. I had originally planned to go after Cyrus for that copper, but Egypt's culture pressing on my cities made me go after them.
Like me, Egypt has no Iron or Copper, so the only offensive units they have are mounted ones. This makes my War Elephants even more mmm, mmm, good than usual. Shaka captures one of Egypt's cities, but Egypt gets it back, so when I finish off Egypt in 1172, I now have 10 cities. I captured 5 Egyptian cities and razed 2.
Tech wise, I learn Education first in 860. I trade it to Ragnar for Philosophy, as Ragnar is pretty low on the score and I figure I can beat him to Liberalism. Halfway to Liberalism, I figure I'm far enough ahead on researching it that I could trade it to Caesar and still get it first. I was wrong and Caesar ended up beating me to it by 1 turn.

After Liberalism, I get Gunpowder then beeline for Military Tradition with the plan of getting cavalry and attacking Cyrus.
In 1274 I finish Military Tradition and start building cavalry. I also start working towards Constitution with the intention of turning ofg research to get the money to upgrade all of my Elephants after I get Democracy. In 1310, Spain goes insane, declares war on me and captures a former Egyptian city I have on their border being pushed by culture. My forces were gearing up on Cyrus's border, so it takes a few turns to get up there, but in 1328, I take the city back with War Elephants.
As is often the case when the AI declares war, Spain had almost no offensive units. She had at most 6 Conquistadors and that was pretty much it. My War Elephants made short work of these and when I finished Constitution in 1334, I went ahead and stopped teching to upgrade all my Elephants. After that, I got Democracy, switched to Universal Sufferage, turned off my reasearch again and started directing all my money towards economic buildings and the war effort.
During the Spanish War, I got Hannibal, who had no negatives with me and was nearly Friendly, to declare war on Spain as well. Sharing war put him over the top to friendly, and I was counting heavily on his vote for my Diplomatic Victory. Ragnar was also on pretty good terms, but the fact that he kept asking me to join his war against Caesar and the fact that Caesar was the only Civ willing to trade me techs forced me to kill that relationship.
In 1388, Spain has one city left and randomly decides to capitulate to Shaka! This brings Shaka into war against me and Hannibal. Seems like pretty bad programming for a Civ to accept a vassal that has one city and is at war with two others, but whatever...
It takes a while to finish off Spain's last city because of dealing with Shaka and the fact that they've upgraded to Riflemen, but it finally happens in 1472, just 20 years before Spain discovers the new world!
Now here's where things turn ugly. After I capture ONE city from Shaka, Shaka randomly capitulates and becomes Hannibal's vassal. I'm not quite sure what's up with this as Shaka still has 4 or 5 cities and I've never been able to convince an AI to capitulate if they had more than 2. Before this, Hannibal had less population than Caesar and Wang Kon and I was counting on their Diplomatic support, but he gained so much population from this that he far surpassed them and I knew there was no chance of them catching up. Looks like I have no Diplomatic Allies!
Anyway, instead of putting me at war with Hannibal, this forces Shaka to make peace with me (without my consent!), and I take my forces back over to prepare for an assault on Cyrus.
At this point in the game, I still don't have copper or iron.
From 1535-1664 I fight Persia until they capitulate with 2 cities left. Sometime during this I turned my science back on and started teching towards UN. I wanted to go Diplomatic and I'm determined to do it no matter what's gone wrong! In 1691 I get Mass Media and start building the UN. I then start working towards tanks and helicopters so I can get some modern war going, since I'm going to have to conquer more population to have enough votes! In 1784, I have all my cavalry upgraded to helicopters and I attack the Zulus. I had a tiny hope that this wouldn't count as me declaring war on Carthage, but no such luck.
The Carthage/Zulu war went from 1784 to 1806 when Carthage capitulated. Overall it went pretty easy with me buying 2-3 tanks every turn. Carthage had a few tanks, but not many and the Zulus didn't have much. Carthage capitulated with 2 cities left. At this point I quickly take out Ragnar as well and then I start an unnecessary war with Caesar in case I needed the popuation and because there was no danger of Caesar's cities having their culture expand before the end of the game.
After I built the UN, no one won the first election and maybe the second. It was between me and Hannibal and we didn't have any outside support. After I started attacking Hannibal, a bad thing happened and the AI's elected Caesar over me for the position. I'm not quite sure why Wang Kon voted for him. They were two different religions at this point.
Anyway, after Caesar's run, I finally manage to win an election while I'm attacking Ragnar/Caesar and in 1819 I'm able to propose a vote on Diplomatic Victory. I win the vote, but between turns my culture expands in some of my cities and triggers a Domination Victory in 1820.
Things that went wrong:
1. Vassal States make Diplomatic Victories harder and have to be handled carefully.
2. When I discovered Wang Kon kinda late, he already had -4 from "You've traded with our worst enemies!" I think it's pretty ridiculous that this can happen before you've met someone, but I guess it's important when going Diplomatic to make sure you get relations with everyone as soon as possible to make sure this doesn't happen. I knew where he was long before I bothered building a boat to go contact him.
3. If I had planned better, perhaps looking up just how long it takes before a new UN election occurs, I maybe could have stalled my war with Hannibal a bit to ensure that some of those cities didn't expand before the election, but I dunno how well that would work.
4. Don't trade Education! Ugh.
Regardless of the reasons, it's quite upsetting to go from an extremely probable award for Diplomatic Victory to a virtually guaranteed lack of anything for Domination when in reality I got both on the same turn.

Maybe I shouldn't have been so stubborn to stick with Diplomatic after Shaka went to Hannibal, but that doesn't make it any better.