Avoid having the USA make friends with Britain. A CSA victory would almost certainly involve a lot of fighting along the USA-Canada border. This would be the third war the USA had fought against Britain. Having the USA join the Central Powers is cliché but it's quite plausible, assuming we get recognizable forms of the Entente and Central Powers. If the pattern of alliances changes so that we don't get recognizable forms of the Entente and Central Powers, then the USA would most likely sympathize with whichever side is against Britain.
Um... okay, this one? I have to contest this. The United States has long enjoyed a kind of 'special relationship' with Great Britain. It would certainly take a blow from Britain getting involved in the conflict, but... I think it would have recovered, largely because in Britain the majority were very, very pro-Union. The British people, they'd probably be furious with their government for having sided with the slave-holding South. Think of it like Vietnam, in a loose kind of way.
Ultimately, I'd say US-British relations would be interesting but ultimately favorable because, well, the government is gonna want to do everything in its power to ensure that they don't have to get involved in another damn conflict between the states and so makes good relations with both North and South.
Okay, that's more my take, and the OP is welcome to do as they please, but I still think they should avoid the cliche of this ruining US-British relations.
Turtledove's timeline had the CSA eventually become Nazi stand-ins, and while I'd say avoid this cliche, I'd also say it's... not that far off from what could have happened. The CSA was founded on the belief black people were inferior and should be enslaved. That's one step away from outright genocide. If they become villainous, don't turn them into outright Nazis, but they should still be heavily racist, because, well...
it's the freaking CSA!
Also, don't just ignore Alaska. Russia now has a very, very expensive colony to deal with they don't have much value for.