I will tell you the single most helpful thing that helped me in constructing characters for a story.
That is the Alignment System.
It is often used in role-playing games to construct broad characters, but I've found it is a great jumping off point for creating a more detailed, well balanced character.
Constructing a 2d grid and plotting good-evil and lawful-chaotic against the axes, and placing characters somewhere on it, it is a lot easier to get a clearer picture about who a character is, and thus what would motivate them.
Having the main 9 denominations is helpful, because it allows you to start constructing a character around the values that each of the denominations holds. However, most characters do not fit perfectly into a single one of the categories, and even fewer stay where they are on a grid as their character develops.
However, everyone fits somewhere on it. A good story will tend to have people across the entire area of the grid.
This can then be expanded to create finer details within the personality of the character, because it's easier to think of how that character would act in a situation based on their values, which is what most people use to make tough decisions. You'll find particular personality traits go with particular alignments as well.
I have difficulty creating characters who are lawful-good or chaotic-evil, because I struggle to find motivations for people who are so virtuous or so immoral, but sometimes it is good to throw a character like that in to advance or halt the story, as they tend to be the most stubborn.
Using this will hopefully allow you to get around the "evil for the sake of evil" problem. Some people have no regard for others, but do not break the rules. Others are rule-breakers, but only in order to help others.
Think of Robin Hood vs the Sheriff of Nottingham. Depending on where you stand personally, either one of these people can be seen as the good guy or bad guy.
As for the difference between protagonist and antagonist, it is very easy using this chart to see which people would naturally oppose and support each other.
A lawful evil and lawful good character will, despite being on other sides of the morality scale, find common ground in respecting the rules. A lawful good and chaotic good would, despite both wanting to help others, disagree strongly on the method of helping people.
People directly opposite on the grid will invariably be the most at odds with each other. Whilst conflicts do not always have to be from entirely opposite ends of the spectrum, like with characters it is a great jumping off point. This can then be adapted to what fits into your story.
If you're still struggling to identify your characters with certain alignments, try looking at some alignment grids of other stories. This is a list of some good ones, including Nintendo characters, The Big Lebowski characters and even the various forms of The Doctor from Doctor Who. But there are a million more if you search for them.
Whilst I have found this as a great help, don't rely solely on this. Characters can become caricatures if they are created in order to fit into this grid pattern, just use it as a guide. And remember, characters can fit into more than one. Batman fits into them all, but then again, it is Batman.