1) War-Mongering: Rome or Boudica; Kublai, Persia, Hatty (with horses)
2) Teching: Liz
3) Culture: HC, Liz
4) WonderSpamming: Any Ind leader; Roosevelt, Ramesses
5) Best early start: HC or, if you eliminate the "exploit" uu, creative leaders like Willem, Cathy, Hatty. Roosevelt is very flexible, so I would vote for him maybe.
6) Best mid-game: Liz/Frederick
7) Best late-game: Liz, USA
8) Pangea Map: Rome
9) Acheopolego: Dutch
10) Continents: Someone who can adapt; Liz, HC, Ramesses, Roosevelt
I would say the best/overpowered leader is the one that can adapt to any situation. For me I think of 3 main situations: Rushing, Rexing, and Coastal start. You could also add isolated start, although I don't play those out myself. So, what civ can rush, rex, or handle coastal starts all together and still do well? HC can rush (quechua) and coastal can go GLH/colossus. Rexing he does ok. Liz doesn't have a great rush or rex. She has an ok coastal opening. She gets stronger as the game goes on imo. These two are my top choices, but I also think that Roosevelt has a lot of adaptability.
Ok, let me break it down more.
To rush, you need a decent uu and/or good opening techs (mining/ag/hunt). Creative helps. I like to have a horse based UU in case I don't have copper. So, Persia, Egypt and to a lesser extent Mongolia, Carthage I like as rushing civs. I especially like Hatty, Kublai, and Cathy has a strong rush too although she may have to rely on conventional chariots. I think the strongest rush though goes to Hatty, especially if horses are in the capital bfc.
Rexing you need to be able to capture land quickly and pay the bills. Creative helps a lot. Here I like Darius, Dutch, Cathy. I think Dutch are the strongest at REXing though. Creative + financial is nice for rexing.
Coastal starts I like Roosevelt a lot. Industrious helps for GLH/Colossus. Organized for cheap lighthouses. He starts with fishing.
I think a leader that does all of these well is tough. Fishing is great for coastal, but bad for rushing/rexing. If I had to pick a leader though that could start inland or on the coast, I would pick someone who had fishing and ag. Food asap is too important. For coastal starts the GLH isn't boosted by any resource, so industrious is really important. Industrious doesn't do a lot for rushing or rexing per se, but if you nab certain wonders along the way, it can help a lot (e.g., oracle->col).
So, ideally I'd want fishing-ag; however, i like starting with hunting for the intel because then you can plan your strategy much faster! Argh!
I think this is why it is so hard to pick someone who shines across the board.
Mining, Ag, Hunt, Fish are the techs I would be ok starting with. Wheel is handy, but not necessary. It's only really good if paired with ag or fish and you get floodplains so you can start with pottery, esp if financial. Myst I don't like because early religion is a gamble on higher levels.
Ind, Phil, Fin, Org, Cre, Spir are the traits I would be ok starting with.
My preference is to play a leader that has min/hunt or ag/hunt and to reload coastal starts

I like to rush or rex. that is my style. land is power.
I've never been able to settle on a leader that satisfies everything I like. I couldn't even custom make one.
If I had to, I'd pick hunt/ag cre/agg with war chariots as my uu. I care much less about my ub. However, this set-up sucks for isolated or coastal starts! You can see here why I like hatty, kublai, darius. I like hunt-cre to get intel and fast setup asap. Good to know if rushing or rexing and where. I like ag for worker-growth first. I like agg in case I rush with axes and war chariots in case i have horses instead of copper.
For coastal, I'd want fish/min ind/org with a later uu. I like Roosey here. Hannibal is ok although getting GLH is tough.
Sorry for rambling!