Like pretty much everywhere, you'll find racism in NZ too. But here the main targets of most racist comments/actions are generally those of Asian, Muslim or Maori (NZ's indigenous people) ethnicities. There have been a few (generally isolated) cases of physical or other severe racist attacks... but these are generally few and far between.... please don't be put off by Tothebatmobile's comments... he makes it sound like we are stuck in 1960's southern USA with his talk of the KKK... it is not at all like that here. Most racism in NZ I think stems from lack of understanding, rather than hatred... although, I must admit I am "white", so perhaps its different from a non-white perspective.
That said, the North Island, and the cities of Auckland and Wellington in particular, are much more multi-cultural than the South Island. Auckland has a large Asian/Indian community as well as many Maori and Pacific Islanders... not too many black Americans, but definitely some. Christchurch (and the South Island as a whole) are very 'white' by comparison, so there is generally a much lesser understanding of other cultures. Big cities are generally more tolerant and understanding than rural areas, but as you've asked about nightlife & dancing, I would DEFINITELY avoid the rural areas!!
Racism against Maori is probably similar to the racism in America against the native Americans... an early history of being considered a "lesser" people by the first white settlers, land & possessions 'bought' (for a few muskets) or just plain stolen off them, has resulted in many Maori being in a lower socio-economic situation than the general NZ population. Pacific Islanders (Samoans, Tongans, etc) are generally treated similar to Maori.
Most other racism in NZ is probably more to do with language and cultural differences, rather than the colour of their skin... e.g. Asian, Indian or Muslim immigrants... again, a lack of understanding and threat of something different, rather than a general hatred or malice... or the fact that their skin is a different colour. A lot of immigrant communities tend to be quite insular, and I think some kiwis are threatened by that, and think that they are not trying to 'fit in' to our culture... which is where the mis-understanding, intolerance & racism comes from.
Given that American culture is very similar to Kiwi culture, I don't think you would have many (if any) problems here... in fact, people will probably hassle you more for being an American in general, than for being African-American... there is quite a lot of anti-Bush sentiment here in NZ (and anti-Iraq war), which many NZ'ers seem to tar every American with, whether they support Bush and the war or not... but again, nothing major enough that should put you off coming here.
I spent 6 weeks travelling around the Southern states of the US a year or so ago... In some places I could feel the tension between "whites" and "blacks"... I have never felt that here in NZ.... racism here is more about innappropriate comments from someone that hasn't thought before they spoke.
In terms of nightlife... I think Auckland & Wellington are probably your best bets... larger cities with a variety of clubs, pubs, bars and other entertainment options. AUckland gets most of the big concerts from overseas acts, but Wellington is more of a "metro" city... great cafe, arty, 'hip' scene. Can't give you much advice on the dance scene, as I'm not really into it, but I would think that AUckland is probably better, just due to sheer numbers of people (its our biggest city, with about a quarter of NZ's total population).
If you're not sure about NZ, I would come for a holiday and travel round for a bit, see how you find it before committing to moving here and going through the rigmarole of getting the appropriate immigration approvals etc. Hopefully you'll find that your fears are unfounded and that we're a pretty friendly bunch in general, whether you are white, black, pink or polka-dotted!!
Best of luck with whereever you end up...
Have lived in NZ all my life - both in rural South Island, in Christchurch and in Auckland, and have travelled the rest of the country extensively.