Personally, I like the thought of an evil paladin who takes one of the 'good' oaths, simply because it adds a nuance to the character. There would have to be a reason why that Paladin took that oath, and not just because "It gives me powers!"
In the real world, everyone has their inner darkness. Some people fight it, some embrace it, but nobody actually believes they are being evil. Evil people believe they are good people. It would be the same for an Evil Paladin.
A Paladin gets his power from his oaths, so he must believe in them. The Oath of the Crown is about following a King of some kind. Believing that the ultimate good comes from following that King. Doesn't matter who the king is. So an Evil King would have Evil Paladins. Even a Good King might have an Evil Paladin to do his dirty work, and that Paladin would be more than happy to do it, because he believes it is a 'good thing'.
The Oath of the Ancients is about beauty and light, and would be harder for an evil paladin, though not impossible, especially if that Evil Paladin is trying to atone for his sins in some way. He would struggle, day and night, to uphold the oaths he took. It would be a struggle for him, fighting against his own nature to be evil, while trying to practice showing mercy and forgiveness. It would create a very compelling character, one that anyone would want to read about, and I would love to play.
Oath of Devotion is a little easier, because the individual can simply focus his 'evilness' on the enemies. He protects the innocent out of duty to his oaths, which is a lawful trait. It's like Dexter the serial killer, only killing bad guys as his way to satisfy his urges while at the same time helping people.
Of course, the Oath of Vengence is the easiest, and even more Dexter-like.
So it can work if you don't view Alignment as a static thing, but more of a current state of mind. An evil character can still be the hero of the story. Look at Artimis Entreri and Jarlaxle. They fight the evil in their hearts, sometimes giving in, sometimes not. It doesn't mean you're playing out of character if you have an evil character show mercy on someone, if it fits with the character. They still love, laugh, form attachments, and the like.
Deciding an 'evil' person has no good aspects is as boring as a good person who has no evil aspects. Everyone struggles with themselves, and everyone has their reasons for doing what they do. I could see an evil paladin choosing any of the oaths.
But as the original question is about how to break their oaths, here's my thoughts: for the Oath of the Crown, the oath is to the crown, so if the paladin, for some reason, no longer believed the Crown worthy, then he would be breaking his oaths. For the Oath of Devotion or Ancients, giving in to his darker side at a pivotal moment would do it as well. For the Oath of Vengance, it's actually easier. See, the paladin will need to choose a 'sworn enemy'. Or enemy type. Similar to the Ranger's favoured enemy. The tenants of the Oath of Vengence say these sworn enemies get no mercy, and the paladin must do everything in their power to destroy them.
Like one of my players chose this, and his paladin hates those who worship evil gods. So what happens if he meets someone, grows attached to them, perhaps even falls in love with them, and then learns that they follow an evil god? What happens if it turns out that she absolutely will not renounce her god? She is his sworn enemy. He is not allowed to show her mercy, nor is he allowed to just ignore her. To do so would be to fall, especially since he had already fallen once. Wow. I just figured out his next adventure. I'm such a d$#%.
