It was really hard for me to pick one. I really do like most of the civs, even among what I consider the "most fun" civs there's not just one. I pretty much dislike only the French (cause they suck), the Zulu (because they are boring, Impi tactics every time) and the Russians (because they have such weak bonuses, still better than the French but there's not really anything the leverage other than the map). I'm also not too keen on the Germans either because like the Zulu they are fairly pigeonholed in how to use them (abusing the unit upgrades) unless you want to ignore their meaningful advantage.
I ending up voting Chinese because they are definitely strong (better than all but the Americans, in fact) but aren't completely broken and I like the way their bonuses synergize so well.
Note that a lot of my opinions might be affected by the fact I only play Deity, random civ.
-Chinese has +1 pop out the gate. Huge advantage. Free Writing means you can get CoL faster than anybody but Rome, free Literacy means you don't have to tech it directly and can still get the 1st research bonus at 5 techs. China is always powerful and a has a crazy headstart on tech so they can do well in any situation or use any general strat (like horserush or oxford bombers) very effectively. Libraries are so cheap so early you can actually justify building them for once (even the Engineering bonus can do it in just 20 turns), further cementing them as the true tech juggernaut of the game, not Japan.
-America, because well, 'murica. They are stupidly OP though, so didn't vote for them. It's straight up impossible to lose as America against the AI even if you don't try...just go gold and unit spam yo. You can win any game, any victory type with the Americans faster than anybody else. That's the power of settler spam at 1/3 cost.
-Mongols are always fun, games are always dynamic because of the hut thing and mountain bonus is nasty as it kicks in right around Knights. 3-move Knights being pumped constantly from any city with a couple mountains can flat out overwhelm the other civs with pure macro power.
-Japan's free harbor bonus from games start is my favorite passive bonus, as any city can grow working only water. I always seem to do much better with almost any other civ though, and the Americans/Chinese absolutely leave them in the dust in terms of teching power. I never even notice the Knight bonus, the anarchy bonus is too late to matter, and while free loyalty is indeed nice, it's not exactly a powerful bonus to be leveraged unless you plan to be attacked a lot before you can make defensive armies...
-Aztecs healing bonus is almost as broken as the American rush bonus, making their horserushes incredibly successful, and any defenders impregnable to mass attacks. Almost as strong... Well, not even close actually, but they don't have much else going for them so it might as well be powerful. The starting gold is good for a walk in but hardly much else besides shaving a few turn exploring, +3 science on a temple is useless as temples are useless (build 2 settlers instead = BAM at least 8 science no matter the era with proper settling, and can grow to more later) and I guess a lot of people don't realize this, but the extra gold bonus actually makes your units and buildings cost more to rush as well, meaning the only thing it does is get milestones faster and makes it harder to rush expensive things like wonders....
-Spanish are really cool for the exploration gold and immediate Whale access, as Navigation is usually kind of a back burner tech for most games (you're more concerned with expansion first). They also have a sick Galleon fleet + Legion strategy that is freaking awesome when you can pull it off. Their other bonuses are kind of meh.
-The English are the best "sit there and turtle, go win with high tech" civ in the game. I mean, even the AI is a notorious turtle. Immediate access to Dye is cool if you can find it to settle on initially, as it makes a horserush or CoL come a few turns faster with even one Dye. Pikemen-strength archers mean you can control the entire map for half the game or more by just spamming cheap units that are extremely hard to dislodge, they have the fastest Knight rush in the game and their double naval support is devastating, especially combined with the Knight rush. Cruiser fleet + Knight armies will take down almost anything, you don't even need to go to tanks for any reason. The English have no weak points in my book, but they can be out teched by better civs.
-Rome starting with CoL means alternate tech openings and way fast expansion (pretty much only the Mongols can get as many cities as early, though all are eclipsed by America later), and the wonder bonus means you might build a lot of them for milestones and actually go for early cultural victory for once. They also have this strat where they spam early cities+roads and just overwhelm with Legions, but it's not exactly as fast or effective as a horserush anyway so...
-I like the Indians pretty much purely for their resource access bonus. It lets you delay getting certain techs (BW, Navigation, Monarchy) just to access useful resources, and can shave turns off horserushes with the right ones. Free Fundy at 5 techs also adds potency to horserushes, but Legions are a bad idea in my experience unless you can also nab Navigation. In which case, you'd have mathematics and catapults already, which are a much better option than legions still. The settler bonus is pretty pointless as all your major expansion should be done by then (you should be nearing Democracy at 14 techs or ending the game in Fundy without expanding much) and at that point anyone can expand further easily anyway (any 4 pop city cna make its own settler in 5 turns). Cheap courthouses are basically useless, like most other buldings...
-I like the Arabs for the fast catapult rush. Then there's the obvious Fundy warrior armies and super-powered horsemen/knight stuff, but those aren't as fun.
-Egypt's okay with Colossus or HGB start. Without either, not so much. Desert bonus is cool and all but unlike Japan, desert is much less abundant than water. The starting wonder and desert bonus are literally the only good things about them, so getting screwed on the wonder half is pretty much like playing Russia. Which is fine, but boring.
-Greeks are kind of oddballs, but can turn into a later game tech powerhouse. The early Democracy keeps you safe and gets you to CoL to expand faster, but they are overall the most conflicted civ which makes them somewhat micromanagement intensive. You have to manage swapping govt. at the right time in the early game, their GP bonus doesn't even work in Democracy unless you are a fool who builds things like temples, the library bonus is nice but later than China's so China has the clear tech advantage with earlier cheap libraries + an extra pop per city +a fast path to Democracy anyway, and the food on water bonus is weaker than Japan's because it comes so late. Etc. Etc.