Lots of studies have shown that when we dislike someone, often it's because on some level we fear we have the same traits as they do. Our conscious mind denies this, of course.
This fact also gives us the key to writing three-dimensional antagonists. I don't remember where I read this phrase, but somebody defined the antagonist as the person the protagonist might have become if one or two things had happened differently.
Another related point is that the Enneagram, a system of defining personality, says that under stress we often become the opposite of our normal self. For instance, one of the types, the Achiever, whose basic drive is to feel valuable, has the virtue of truthfulness and authenticity, but under enough stress will be deceitful. (I know I'm oversimplifying this--if you want to find out about it in detail, and how to apply it to writing, check out The Literary Enneagram: Characters from the Inside Out, by Judith Searle.)
If we put all this together, it suggests that a good way to craft a strong antagonist is to look at the basic traits of your protagonist, think about how stress or negative circumstances could twist those and drive him or her to the opposite end of the scale. That will give you a back story for the antagonist.
A good example of this is a character I created for a post warning about the kinds of people who can make a writing group a less than perfect experience. This is how I portrayed him (they also come in a female version):
JEKYLL AND HYDE
This writer is the most pleasant person you can imagine. He’s happy to share stories with you about how unfair agents and publishers are to writers like you and him, who surely deserve to get published. No, put that money away, he’s getting the drinks…until you get something published. Then he’ll turn on you in an instant. If he still buys you a drink it’ll be only because he’s slipped a little cyanide in it.
What causes this characters to turn is that he thinks he has everything in common with you but when you get published he feels almost like you've betrayed him. Misery demands company!
What makes him a good character--at least for readers who are also writers--is the fact that most of us wonder whether we could become bitter and malicious if we had enough rejections. That makes it hard to think of him as totally an "other", a person whom we can dismiss because he's so different from us.
Of course this works with non-writer characters, too, because it's not really about writing, it's about jealousy, fear of rejection, anger at rejection, feelings of abandonment, all things anybody can relate to to some degree.
If the antagonist in your book or screenplay doesn't have enough depth, take some time to think about what he or she has in common with your protagonist and figure out how and why things went sour and turned the antagonist to the opposite, to the dark side. You don't have to reveal all that information but knowing it will help you craft a worhty opponent for your protagonist.
(More writing tips in Your Writing Coach and Your Creative Writing Masterclass, both published by Nicholas Brealey and available from Amazon or your other favorite book seller.)