Personally I don't think this list is fair due to the terrain. The terrain plays a H.U.G.E. role in who's good and bad in each particular game. Now, if you want, use a static map, like Earth 18. I'll do that, from my experiences of the map and from watching the King of the World series:
1. Qin Shi Huang
This guy, using his superior land, will out tech, out build, out grow nearly everyone 99% of the time. On VERY rare occasions (aka that 1%), he might get boxed in a bit by a combination of Genghis Khan and Tokugawa. That is, if Tokugawa somehow gets a foothold in Asia, which has happened. Other than that, mid-late game this guy just sky-rockets past everyone in everything. Often vassalizes Genghis, sometimes Tokugawa.
2. Cyrus
Oddly enough, even though he's using some incredibly crappy land (The stuff north of Iran.. mostly desert / plains with some seas), he can do very well. His main capitol is a production powerhouse, which probably helps things along a bit. Often gets a vassal or two, usually Saladin or Alexander, on rare occasions Asoka.
3. Catherine
Using the vast expanses of Russia (well, eastern Russia), Catherine can make a pretty strong empire. Often is a military and tech powerhouse and will war with her neighbors if they aren't of the same religious bloc. Though she rarely gets vassals, and can be manipulated into attacking a neighbor fairly easily.
4. Frederick
Often competing with land with Catherine, Freddy most often becomes one of the more dangerous people in Eurasia. He's never a weakling when it comes to military, and often has a strong economy to back it. Often gets a few European vassals like Louis, Julius, or Alex.
5. Montezuma
Nearly always #1 in the military, Monty's primary short-coming is his lack of naval intelligence and his isolationism. He always steamrolls over Roosevelt, and if you remove the Mesoamerican mountain in panama, he'll often vassalize Capac, sometimes colonizing South America too. Never a late-game powerhouse though, after the middle ages he usually just sits around, building large armies of obsolete units and twiddling his thumbs - or waiting for you to put a colony on his continent, in which he sends mountains of catapults and macemen to throw themselves at your superior, but vastly outnumbered units.
6. Hatshepsut
Hatty starts out with amazing cottage-able land. 3 strong cottage cities on flood plains if played correctly, though she usually only makes 2. Her primary cities are low on production though, so often her military is weak until later. She has a habit of settling cities in the middle of the Sahara, just to fill up space.. but often she can keep up with the tech rate of the Old World.
7. Asoka
While he's often a peacemonger, I've seen him build a large army that rivaled some of the more militaristic civs. Thing is, he only uses it in defense. Often he's the founder of a religion and enjoys spreading it to the rest of Asia (sometimes the middle east), creating a religion bloc there. Quite good at teching and economy as well.
8. Mansa Musa
Just like in custom games, he's a big techer. You can always count on him to have good technologies at affordable prices. Often vassalizes to Hatshepsut though, since he doesn't spend much on military. He also has to contend with the barbarian hordes of Africa.
9. Elizabeth
Another great techer, though she's content to live in Britannia. Sometimes she'll lose Ireland, or sometimes even Scotland to the French, on rare occasions Germany. After that though, she usually doesn't expand until Astronomy, when she often gets Iceland and sometimes Newfoundland.
10. Hyuana Capac
10+ is really where the remaining civs all do poorly in games. I chose Capac for the highest of the remaining 8 since if allowed to expand, he can often encompass all of South America and build a strong, if still behind economy. Never has to worry about wars for the most part, since Monty rarely goes into SA. All he has to worry about is the barbarians. Now, if he isn't allowed out of his prison.. he only builds 3 fairly medium sized cities and is often far behind in everything. On RARE occasions he'll found a religion and spread it to the rest of the Americas.
11. Genghis Khan
While usually ends up vassalizing to Qin Shi Huang, on rare occasions, if he's aggressive at the start, he can stump the growth of his southern neighbors, and not vassalize until end game. Nearly always controlls most of Siberia, which is crappy land. Often has a large, obsolete army and a poor economy.
12. Tokugawa
Is fiercely independent, like most games. On rare occasion he'll submit to Qin Shi Huang, but usually just sits in his little corner of the world doing nothing. He'll sometimes expand to parts of Indonesia, and even on rare circumstances take Hawaii. Even rarer will he get a foothold in China and/or Korea. Once he did that, and actually was one of the top competitors in the game, knocking off Qin as a competitor. Only once though. Occasionally he will colonize Austraila too.
13. Isabella
Ah, our religious fanatic. 99% of the time she'll found a religion, and end up making Europe one big happy family. I've seen her have 3 holy cities just on the Iberian Penninsula. Usually has a strong, if very small economy and a weak military. Not much of a threat, more of a prize worth stealing, especially if she has shrines in all her holy cities.
14. Saladin
Another religious fanatic, usually founds one religion and spreads it throughout the Middle east and Africa. Other than that though, he doesn't do a whole lot, and often vassalizes to Cyrus or Hatshepsut. Sometimes colonizes Madagasgar though.
15. Louis XIV
The wonder-spammer. He'll always get the first few wonders, which is wierd since the Great Wall is useless in Europe. Sometimes he'll try and colonize Ireland, Scotland, and Iceland with varying degrees of success. Other than that though, not a threat.
16. Alexander
Ah, Alex, very dangerous person in regular games, an oddball on Earth. He's content to stay in his little island chain of Greece, only rarely leaving to Turkey. Usually vassalizes to someone bigger, often Freddy. On one occasion, in one of the KotW series, he was able to heavily attack Hatshepsut, only to be beaten back and eventually eaten by his neighbors.
17. Julius Caesar
Pretty much on par with Alex, he'll rarely get more than a second city before being blocked in. On one or two occasions though, I saw him attack and even kill (mostly) Freddy, taking Freddy's place of power. Rarely happens though, instead he usually just vassalizes to a stronger neighbor.
18. Roosevelt.
Ah, Roosevelt, poor little Roosevelt. Rarely survives past the middle ages without getting sacrificed on an Aztec altar. If he does survive, theres a 98% chance he'll vassalize to Monty. Never is he a threat in any game. Poor guy, he doesn't know how Americans run things it seems!
So, there you have it, a list of best and worst AI on the Earth 18 Map from my experiences. These may change according to what the Player does of course, but thats always bound to happen.