Great idea for a thread. I'm still kind of disappointed by how poorly the G&K AIs perform overall, so I'm hoping the BNW Civs will be a little better at playing the game.
Below is a summary, in vaguely strongest-to-weakest order, of how I think they will perform, with some rationalisation of my thoughts. Note that this is based off how the AI in G&K tends to behave in single-player King games, so I'm making a pretty big assumption that AI behaviour hasn't been tweaked at all. Things like Shaka and Ashurbanipal being aggressive should be givens, though.
Likely to be strong
The Zulu I think will be the biggest threat. At first I was reading them in the Montezuma-mould of a fail war-based AI, but I think they might be more like Germany, especially with the lower maintenance cost. Screenshots and reviews so far have shown that Shaka likes to attack City-States, which won't make him popular, so if he is too aggressive too early, he might get dogpiled and wiped out - a similar fate sometimes befalls Germany in my games.
Another strong performer should be
Poland, with their seven free Social Policies and a pretty powerful-looking UU. We still don't really know exactly what role Social Policies will play, but since cherry-picking Policies does seem to be the way that part of the game is now played (and the AI loves to do that even now), I would expect to see Poland going for a lot of the Tree-opener Wonders. Casimir should also be able to develop his Ideology faster than other players, but I don't see him having much interest in culture or tourism, so it'll be interesting to see how that plays out. He will probably like attacking his neighbours and going wide.
Assyria will be aggressive for sure, but a big part of how successful Ashurbanipal is will come down to whether he's aggressive against the right people (i.e. tech leaders). The Siege Tower suggests he will want to go to war early, and he should be a pro at capturing cities (insofar as the AI ever is). I can see him getting an early advantage and being able to take out his weaker neighbours by the Medieval Era - hopefully picking up some useful techs along the way. He could struggle in later eras though if his warmongering upsets other AIs.
Morocco should also do OK, seeing as their bonuses are largely passive (although part of this depends on how well the AI handles Trade Routes). I predict a relatively non-aggressive focus for the Civ, given the reliance on trade and defence, but they might get attacked by their neighbours. I don't think they will go super-wide, but their desert start bias might lead to some big cities (Salt, Flood Plains), and Ahmad will probably love to build Petra. They'll also be a pain in the bum to conquer, especially if they have Berber Cavalry sitting on Kasbahs (but then, does anyone trust an AI to put its units in the right place?

).
Could go either way
The Shoshone will be land-grabbers and city-spammers extraordinaire, I expect, and presumably their defence bonus in friendly territory will make them hard to defeat. I think Pocatello will be an annoying opponent for the human player, although perhaps not a formidable one. I can see the AI being completely inept at using the Pathfinders to best effect - although they could be scary in the early game as neighbours, I think they might peter out in the mid-to-late game.
Portugal is an interesting one - on paper, they will be one of the most powerful new Civs in the player's hands, but I think they AI will stuff them up. The AI will most likely be useless at getting Feitorias built, so there goes one of their most powerful traits, but on the other hand, with Mare Clausum and the Nau, Maria won't have to work very hard to earn money, and wealthier AIs are usually the best performers (see Pachacuti and Harun). Again a lot of this depends on how good the AI will be at working with the new trade mechanics. At this point, since it rarely seems to struggle for money anyway, I don't believe that it can be trusted not to stick with domestic trade routes and wage war on its neighbours instead of trading with them. I do hope I'm wrong, however.
I haven't seen much discussion of
New France here. I don't think we can expect Napoleon's personality to change very much (he is firmly in what I like to call the 'slicey' category at the moment), but I wouldn't be surprised if he ended up performing less well than at the moment. Assuming that Ancien Régime's +2 culture per city is going, his new UA does seem like a downgrade in AI terms (I don't really believe that the AI will be very good at theming its Great Works). We don't know enough about the Château to comment on how that will impact his game, but on the face of it, he's losing his easier culture acquisition from the start of the game (not to mention the Foreign Legion) and gaining something that, while nice from a human player POV, may not benefit him a great deal. Judgement reserved for now.
Brazil I am classing as mediocre-to-weak at the moment. They're clearly the cultural powerhouse of the expansion, and with the AI's happiness bonuses Pedro should be swimming in Great Artists, and should be able to bag a few Wonders as well. They should also be pretty resilient to outside Ideological pressures late-game. On the face of it, he could be the new Darius. So why the uncertainty? Well, partly the fact that AI in CiV generally doesn't have a great track record at managing Great People or timing Golden Ages well. Nor does Pedro's Carnival benefit his military in the way that Darius' does - the UU is so late as to be all but irrelevant in military terms. We have already seen some evidence of Pedro getting attacked and losing cities in screenshots from preview builds (OK, not the final game, but still) and I think he might struggle to stay alive.
Likely to be awful
For these, I don't much disagree with other posters' comments.
Venice are almost guaranteed to be useless in the hands of the AI. Enrico will be forced to go on the offensive from the get-go, but apart from the Galleass, he gets no particular advantage towards capturing cities. I hope he at least has been programmed to appreciate the particular usefulness of the Merchant of Venice, and doesn't instead spam Custom Houses everywhere... but I'm not sure I believe that he will. He will probably be hyper-aggressive, spamming units when others are building Settlers, and unless he wins those early wars he will disappear fast.
Indonesia will also do badly, but then that's hardly a surprise - both Venice and Indonesia feel very much as though they have been designed to provide a fun and challenging playing experience for the human player rather than as challenging AI opponents. The AI most likely won't understand that it needs to build 3 cities on a different continent to get the bonus from its UA, and the Kris Swordsman is wildly unpredictable. Gajah Mada will most likely have an expansionistic and faith-favouring personality in an attempt to counterbalance the UA problem (if he builds enough cities, eventually some will be on different continents, right?) and this will lead to conflicts with other Civs. The Candi's faith boost will only really come into play late-game when there are lots of religions floating around.
As I said at the beginning, I'm making big assumptions regarding the state of the AI come BNW's release - so there's potential for most of this to be completely off the mark