Best isn't disputed much I think. Due to an almost brokenly amazing UU, a good UB, 1/2 good starting techs and an amazing trait combo Huayna Capac is above everybody by a bit, though Gandhi and Darius are close on his heels.
My personal favorites are Roosevelt, Zara, Darius, and Gilgamesh.
I'm a newer player (~6 months) and I really like the Creative and Organized traits, ORG being my favorite. It's not that I don't understand or see the great power of the IND/PHI traits or the general usefulness of FIN/SPI, I just like the passive help with the economy and am a fan of courthouse spamming since it helps a ton with expenses. Civic cost reduction+cheap, 2 pop whip courthouses is like a dream to cut expenses and run your slider higher! Cheap courthouses also help with quickly making captured cities in war profitable, and war is oh so important in this game.
Creative is similar in that is helps passively a bit, though its effects totally disappear after the early game phase is over. However, the early game is very important, and the hassle it saves is nice. Never having to fret over your border pops and avoid Mysticism for a long time is nice, the cheap libraries help you tech in a crash or start on that Great Scientist earlier. The quick border pops also help claim land and counter other CRE civs who are frickin' annoying in that regard
America doesn't have anything useful in the UU/UB department since they're both hella late, but Rosie has the coveted ORG/IND combo for a builder. He also starts with Agriculture (good tech) and Fishing (situational, but can be good) which means he's adaptable to any start, being able to get all 3 of AH, Agri, and Fishing if needed before his worker pops and can crank any of the early wonders easily. He and Augustus are actually the two best civs for getting the Great Lighthouse ASAP because of their Fishing+IND start; Rosie has the better secondary trait that speeds up his Lighthouse to a 1 pop whip ( and IMP sucks

) while AC starts with Mining for a fast track to Masonry and BW for chopping it. Heavy abuse of the IND trait for fail gold is always good and a powerful tactic, and you can put that gold to more use in his economy with ORG's cost reduction benefits. Even Oracling CoL to gain courthouses earlier is a viable option for Rosie, to create a roundabout, but solid early economy. Huayna and Darius are better coastal strat civs because of FIN, but I prefer ORG anyway for the empire-wide benefits to go along with the GLH or any other wonder focus, including fail-gold use.
Zara has my favorite trait combo of CRE/ORG. You don't have to worry with popping borders, you can set up library economy faster to offset early war/REX taking your slider down, ORG is well...I explained it already, I like the cheaper expenses. His tech combo of Hunting/Mining is also solid; I like starting with a scout just to see the lay of the land as fast as I can (habit from Civ:Revolution and various RTS games: explore the map ASAP!) and it lowers the cost of AH if his fast BW doesn't reveal bronze. The Stele is meh (it only gives him +3.75 culture for a border war compared to +3 for a normal CRE civ with a monument) but the Oromo is pretty good if you rush to Gunpowder with the idea to get Steel fast and attack with something better than maces before rifles. Drill 4 Oromos beat up longbows and are definitely top dog of the Musket UUs. Haven't defended with them much but they out to be impenetrable to non-mounted with all those first strikes and resistance to collateral. He's just a solid leader for packed or close-in land heavy maps without much focus on the sea like Pangaea, and easy to use because of that trait combo.
Darius has the strongest economy in the game with FIN to boost his commerce and ORG to reduce his expenses. The Immortal can handle Archers pretty easily if you shut down the enemy's spear production (I HATE Pacal man) and it is even stronger and faster at doing what Quechas do, but it needs horses. His UB is forgettable, +2 health Grocer? Big whoop. It'd be better if it was a Market instead so it didn't require Guilds to get it. Starting with Hunting/Agri and being FIN means he has the potentially fastest AH of anybody to look for those horses and he gets a scout to check the area out, which I really like personally. Agriculture and an incentive to tech AH first means you'll have your food techs early, or you can opt to go for coastal: ORG cheapens his Lighthouses and he's FIN at the same time, so he can immediately start pulling in profit from water cities after a 1 pop whip and just working the sea. This kind of flexibility with his economic security and a great, if situational UU make Darius an amazing leader with a powerful early game and follow through as an amazing techer. I actually stopped playing him after a few games because he makes it too easy to avoid learning from hardships!
Ol' Giggles. He's like early game easy mode. CRE and the Vulture, one of the best UUs, let you spread out and take a chunk of terrain through rapid expansion or force. His protective trait means it's easier to hold on to it with simple archers too, if that matters. I have done the hill+wall+CG archer fortress city thing a couple of time after pissing off AIs early, it works pretty well, especially when they have to slog extra far through your borders. That UB is great and gives more reason than the Oracle to go Priesthood, helping to alleviate the economic stress of your early expanses. Faster libraries AND earlier, cheaper courthouses? It's like impossible to kill this dude's teching. And again, that Vulture is badass. 6 power, available with just BW, and you can give it CR to further help it bust archers heads. It even does better than swords against other axes; even if they still beat it, it has better odds in the field (especially defending).