Unholy Mistress
A couple of people have replied and reasonably accurately probably but a hell of a lot depends on the details. For instance how does the south win? There's a world of difference between a minimal war, say because the south wins a couple of early battles and say wins early foreign recognition to one where it out-lasts the north in a war of exhaustion. Again if there is foreign military intervention, most commonly seen by the Trent Incident leading to a conflict with Britain, there's a world of difference between a short conflict after which the US realises it needs to make peace or one where it fights for 2-3 years and probably gets it's economy fairly well smashed.
All those affect the views and behaviours of both nations and others after the conflict. A long and bitter conflict drains both powers and probably means their also maintaining much larger armies for a while afterwards. Some TLs after a war involving Britain has the US trying to compete with them as well which means the north will be highly militarised and probably seriously retarded economically and socially as a result. However I think that's fairly unlikely.
If the war ended fairly quickly then slavery would probably survive in the north for a little while, presuming some slave states remain loyal. Since Lincoln would have no basis for stamping it out and there would be a fear that if he did then any such states might look to leave, possibly joining the south. However the survival of the institution would be pretty short as moral and economic feeling would be again it and probably it would be largely or totally gone within a decade.
However, as said, it would probably be pretty unpleasant being black in the north. Apart from the racism and fear of black competition there might well be some blaming the blacks for the war. On the other hand liberals would be arguing for supporting blacks against the south and there could be a lot of tension about continued slave escapes from the south.
It would still be markedly better than being a black in the south.

Slavery will probably last a long while here, simply because it will be an icon of the identity of the south. You may well see some states moving to abolish or restrict it but again that leaves the question of what happens to freed blacks? It may be that the CSA might try a repatriation programme, possibly establishing their own version of Liberia in Africa.
In terms of economics the south would be primary agricultural, with some industry and other raw materials, although when oil starts becoming significant, if the CSA can last long enough that could be important. Unless it chances it's attitude on central government, which it might over time or a long independence war, it;s going to be highly dependent on imported industrial goods from Britain and probably later Germany.
The north would be markedly weaker without the south and it's resources but not greatly so unless it goes off the rails totally. [Could get some highly militarised state that falls into autocracy and economic backwardness but that is highly unlikely. More likely it would be a bit smaller and weaker than OTL but not greatly so.
The south is unlikely to attack any neighbouring states, such as Mexico or Cuba. Both because after a long war it would have little appetite for it and because neither Britain nor the US would tolerate such moves. With the CSA between Cuba and the north the Spanish-American war is likely to be butterflied. Possibly also an isthmus canal although as the Pacific coast of the US develops there will be increasing arguments for this.
Whether either states gets more involved in international politics would again depend on circumstances. The south might want allies against a new attack from the north but both slavery and the greater economic power of the north are likely to make it an unpopular partner. Whether the north will want to probably will depend on whether there has been a significant foreign involvement in the civil war. If so, almost certainly with Britain and/or France, then it would depend on how long the conflict lasted and the US reaction after their possibly heavy defeat. A period of prolonged tension with both Britain/Canada and the south might make the north seek foreign allies but it might struggle to find them unless it's also willing to make commitments, which could be politically difficult.
I think it very unlikely that you would have anyone that similar to Hitler in this TL and you could well have no WWI either. So much depends on the circumstances.
Steve
Let's say the Confederacy wins by some miracle, how does history take it
s course, I have a few questions that I would like to discuss with all of you
1. CSA's relations with the rest of the world
2. USA's rellations with the rest of the world
3. Civil Rights (Does the CSA ever abolish slavery and segregration, and give blacks the same light and rights as the white man, and what about the USA, without MLK, does the Civil Right's movement take longer?)
4. Does CSA-USA relations ever get to a very good place (ala Canada-USA?)
5. The CSA's impact on wars and foriegn affairs, would the CSA side with the Nazi's? and etc
thanks, and let's have some good discussion on this