They would have elected Grover Cleveland and Woodrow Wilson, as president. Blacks wouldnt have been kept enslaved, but their rights to a public defender and Miranda rights wouldnt have been galvanized. It would have been same as South Africa apartheid. And lynchings, maybe even to this day,, sparsely would have kept going on, due to no blacks having public defenders when whistling at females.
It's hard to say what individual people would have later become President of the CSA. I'm sure many would have been Civil War Vets and/or prominent aristocrats, but that's about it.
Also, everyone born after the Civil War would have been butterflied out. I'm not sure if there's a more specific term for this, but because a person's identity depends on a specific sperm meeting with a specific egg, any change in a person's series of movements before they do the deed would result in a different child being born. For instance, let's say a man goes home from work to engage in marital congress with his wife, producing a son nine months later. If you delay his commute home by just one second, through something as miniscule as the cars in front of him just taking a second longer to get out of the way, his specific combination of movements up to his wife is changed, which would most likely lead to a different sperm, which would lead to a different child - maybe a daughter, maybe a son with more athletic potential, maybe a son with a mental disorder, et cetera.
This means that every President from Calvin Coolidge up to Donald Trump would almost certainly not have been born. Chances are that Warren Harding isn't born either, being born November of 1865 (a victory for the South would have likely meant that the prior conditions would lead to him being butterflied out). Of course, this doesn't just extend to Presidents, but to almost everyone worldwide.
Anyway, the reason I mention all of that is because some of these alternate timelines involve people who would have been butterflied out. There is this funny alternate history movie on what would have happened if the South had not only won the Civil War, but conquered the North in the process. IIRC they had John F. Kennedy winning the Presidency as one of the first Presidents from the North in a hundred years, despite the fact that he would have been butterflied out along with everyone else who hadn't already been born before the end of the Civil War.
So, unless you were to just handwave that idea away, all you can really work with when it comes to alternative history are societal trends, geographical resources, established military/political/economic systems, etc. For instance, the South right after the Civil War would still be an agricultural economy heavily reliant on slavery. Eventually, abolitionism would take root as advances in technology rendered slavery obsolete. This would lead to the question of "What would they do with all the slaves?" Free them but relegate them to second class citizens until a Civil Rights Movement starts? Deport them to Africa?
Their relations with the North would also be a major topic. Would they conflict over western territory? Possibly, though I'm sure a peace treaty would prevent war between the two nations for at least a couple decades. Would they be cordial? Northerners would likely be bitter at their defeat. How would people traveling from the USA to the CSA be treated and vice versa (after all, there would be people with family in both sides)? Would there be another war between the USA and the CSA eventually? It's a definite possibility, especially after a few decades have passed.
What would the new political factions in the CSA be? After all, there's undoubtedly quite a difference between Texas and Virginia. There would still be conflicts between poor whites and rich whites, and some conflict between urban and rural areas.
What would the new political factions in the USA be? There's quite a difference between places like New York and Indiana. There would still be issues with immigration, urban vs rural, how blacks are treated, unions, trust busting, women's rights, etc.
What would their attitude towards Mexico be? If a Great War erupted in Europe, who would they align with? What would happen to the lands currently occupied by Arizona, Utah, New Mexico, and Colorado? Would the CSA take them? Does the Philippines get taken from Spain? Are there any colonial ambitions from either of the two nations?
There are so many questions to answer, and each question raises more answers in turn.
