Diplomatic Victory in 1948, score 25316.
The victory
The Start
I started by moving my settler 1 square West (did anyone not do that???) and going for the classic CS slingshot which I got in 1360BC.
Civil Service here I come...
I didn't follow through the slingshot by heading for maces because it was pretty apparent by then there was noone nearby to conquer, and plenty of free land up for grabs. Peaceful expansion for seemed to make more sense than walking to the other side of the continent to take out the Americans.
At this point I was pretty unfocused, not really sure what victory condition to go for. I filled up the available land then declared war on America to get that land, leaving him with a couple of cities. Beyond that I just expanded while keeping my economy in good shape, and gradually making contact with the other civs. I thought seriously about trying to go for a diplomatic victory at this point, but somehow didn't seem to be managing to get close friends with anyone, so mentally ruled out that option. However, it was obvious I was slowly pulling ahead of everyone in techs, thanks to good economic management and of course that early civil service.
The Iron
About 1600AD, crunch time came, and I have to admit it was caused entirely by a couple of misjugements on my part. I discovered steel, and prepared to build cannons so I could expand again. Ooops, why can't I build a cannon? Ah. I have no iron. I'd actually got to 1600AD not noticing I had no iron (CS slingshot and eventual macemen meant I never needed to build swordsmen). Quick check revealed that America still had control over the only iron on our continent.
The Iron problem
So I declared war and prepared to remove him from the continent (albeit with cavalry and catapults! Grrr!) Then my 2nd mistake. I crushed Atlanta and decided to raze the city so I could rebuild 1 square south of it. Trouble is, I didn't have a settler ready. And of course, 8ish turns later my settler turns up to find a new English city there. I finally get that iron in the mid 1700's. And a new 'You declared war on us' enemy. My likelihood of a diplomatic victory is kinda - ummm, shall we say, not going up. (Ainwood: Why isn't there a special award for going so long without iron and still winning a monarch level GOTM?

)
The Age of Wars
1800ish I declare on Toku to grab his main island. His two cities there are surprisingly well-defended and put up a long fight. Round about this time I picked up oil, transports and destroyers. Noone else was anywhere near having those, and on an archipelago map that was irresistable: I could walk over everyone elses navies (well, perhaps sail over?). Domination victory starts to look very attractive. Even better - by now I have an absolutely unassailable tech lead over everyone else.
I love a tech lead! (This was in 1808)
So mid 1800s I declared war on America yet again, to take out his last cities (on islands to the North). I do that with ease: I have infantry and cannon against longbows and pikemen! The Americans had a defensive pack with Victoria so I got a free war with her too! Victoria proved a
lot tougher: that's partly because she had riflemen, and partly because she was soooo far away, and spread out across the islands in the NE of the world map, whereas my production/troop powerhouses were all near the starting location in the SW of the world. It took till about 1910 to reduce her to, basically, her main island, before I was forced to abandon the war due to extreme war-weariness. Yes, I mean extreme: My core cities were all starving, production was dropping appallingly, etc. And the trouble was, I didn't see changing civics to police state as much of an option because I was relying on democracy to allow me to buy theatres and libraries, then granaries and lighthouses etc. in all my newly captured cities to try and get my land area and pop up. Most of these cities were on islands and had next to zero production, so I'd have been stuck without democracy.
I was also starting to really see the problem with domination on this map. I was by far and away the biggest civ, yet at the end of the English war I had 47% pop but still only still 35% land area. Getting to 64% on this map was going to be very tough. Even if I took out the next biggest person (Isabella, 14% pop, 14% land, I'd be nowhere near. Long slog ahead)
The up side was by now I had panzers. Yay! Fun with panzers ahead...
Isabella's Humiliation
I decided to take out Toku and Issy next, because they were both hostile towards me and both had weak navies defended by frigates or galleons (Russia and Persia by now both had destroyers). And this was
the best bit, the pride, of my game. Of course it was a walkover. Panzers and artillery against riflemen and even longbows. I declared on Toku (around 1920, I forget the exact date) and then on Isabella the next turn. Simultaneous wars.
In 10 turns I took 4 of Toku's 5 cities. His new capital only survived by virtue of (a) being on the other side of the world and (b) Extreme war weariness hit me again so after 10 turns I was desparate for peace.
Simultaneously, in 11 turns I captured 11 of Isabella's 17 cities. I razed a further two, and when after those 11 turns I asked her what the price for peace was, she offered me another city! 17 cities down to 3 in 11 turns. I savour the memory

Interestingly, and slightly counter-intuitively, I think the archipelago map speeded up that war, because my transports could ferry the troops between Issy's cities far quicker than they could've marched through enemy territory.
btw, during the wars, this really amused me. Wot my SAM infantry guy found while out looking for Spaniards. It's not just how late the barbs are thriving, it's where they've put their city!
Modern barbs
Domination to Diplomatic
Now, what to do about this domination? Sure, I was between 40 and 50% land area now, and also well over 50% population. I could get my land area up a fair bit more by settling all the useless icy bits that noone had touched yet (and I was proceeding to do that), but I didn't think I could get to 64% without another war - and war weariness meant that'd be tough going. But with my population, why didn't I just vote myself a diplomatic victory? Durrr, major brain disfunction here, why on Earth hadn't I thought of that before? So I change my tech priorities and start beelining for the UN, which I built in 1940 I think.
1944 - my first diplo win vote. Aaaaaaaargh. Out of 800ish votes, I'm 4 votes short! Noone else votes for me of course, I've long since burnt all my bridges diplomatically.
I ponder. Can I cope with another war? Victoria would be the obvious choice. Probably I only need to take one of her cities to secure victory. And thanks to extreme cultural pressure in Liverpool (the city had actually had an English revolt, I hadn't realized it was possible for a captured city to revolt to its former owners), half my army was now there, perched on the English borders. So an English war was very tempting. OTOH can I get my population up enough for a diplo win without a war? Given I have so many new size 1 and 2 cities, my pop probably is growing faster than anyone elses, so it's possible.
Liverpool under pressure
I decide on my plan.
Firstly, I go through my cities, making sure every single one emphasizes population growth. I start buying lighthouses and granaries etc. in all my new cities. And I even start buying settlers to populate very last bit of outlying land I can find, just to try and get my population up to win a vote. (I did draw the line at settling the gaps between my existing cities though. I have some sense of decency

). But as a reserve, I started moving transports etc. to Liverpool to prepare for an invasion of England if necessary.
In the event the invasion wasn't required. My plan worked, and in 1948, on my second attempt, I was able to vote myself a diplomatic victory quite comfortably. Interestingly, i won with 63% population and 55% land area, so I'm pretty sure I could by now have got a domination victory without too much trouble if I'd chosen.
The Graphs
Thought I'd post my GNP graph because it very clearly shows the effect of war weariness. It had a
huge impact on my GNP. (The high point was a golden age, timed to get me the UN and a few other wonders as quickly as possible)
GNP
And here's the (rather more boring) score graph
Score