Your antagonist needs to have a very clear motif. This is different for evil villains than it is for sympathetic antagonists, and I’ll explain the two separately. I’ll do the pure evil one first because it's easier and will take less time.
A pure evil villain’s motif is always completely selfish. Money, power, world domination, or just plain old enjoying causing others pain. A good example would be Cruella de Vil from 101 Dalmatians. She simply just wants to have a fur coat made from the skin of puppies. Now, this might seem harmless enough, as lots of people in real life wear clothes made from animal hide, but what makes her evil in this context is the fact that the dogs are the main characters. Although in a lot of contexts, she would simply be seen as a fashion designer, in this particular context, she’s willing to kill the main characters for her own personal gain. That is where she makes her way up to a truly evil villain.
Another case of evil villainy is the large, faceless corporation. Both Wayland Utani in the Alien franchise, and Ingen in the Jurassic Park franchise have this core trait. Weyland Yutani wants to bring an alien to Earth, generally either for research or military purposes, not caring about the millions of lives they would kill. Ingen has a similar problem with how they are willing to either weaponize or display dinosaurs in Jurassic Park.
Learning from this example, we can see that the true motif of an evil villain is always in some way personal gain. The reason why they want that personal gain can be vaguely explained, or not explained at all. Although this is often seen as weak writing, in this case it isn’t. With a sympathetic antagonist, you need motivation that’s explained. A purely evil villain can just have a throw away line explain their motif and that’s it.
This is often the reason it's more challenging to write a sympathetic antagonist. There’s often a statement of how the most interesting villains are the ones who think they’re the hero, and this is often very true. Going back to Jurassic Park, its founder John Hammond was simply oblivious to its dangers, and in the end of the movie realized his mistake. I wouldn’t even call him a villain in his appearance. It wasn’t until its sequels when Ingen actually became evil, knowingly putting millions at risk for the sake of a dinosaur zoo.
Zemo in Captain America: Civil War is also a sympathetic villain to an extent. He had his family killed in an event that was the fault of Tony Stark, and thought the world would be better off without the Avengers. He didn’t want to actually take over the world, but he saw the good guys as causing more problems than they solved. Sure, he was wrong about it, and he was misguided, but he had a focused agenda. He also came back as a bit of a protagonist in Falcon and the Winter Soldier, which adds to his credibility as a sympathetic antagonist.
No matter which type of antagonist you’re writing, don’t make the plan too complicated or contradictory. This can often take audiences out, as it takes 1000 coincidences in order for their plan to world, or can simply confuse the audience. Keep their plan simple.
