In writing, as in life, it's often the similarities of things that bring about the most conflict. Familiarity breeds contempt. Here are some ideas to use those similarities as well as differences to enhance characterization and conflict.
1. Is there a particular type of feud that underlies part of your character's background? For example, is her brother in the Army, and her hero's brother is in the Navy, and she meets her hero at halftime during the Army-Navy game? Imagine they are both wearing sweatshirts with Army or Navy on them, and end up stepping forward at the same time in the concession stand. Turns out they have the same order (beer and hot dogs), and even both want only mustard on their dogs. How might the similarities as well as the differences fuel the conflicts, yet bring them closer together?
2. Write a backstory for each major character, so he has a reason for being the way he/she is. Did your hero become an outlaw because he broke the law while saving someone's life, but he hides this? Why? Does he have a dark secret about his past? Who else knows about it? Could this come out at a worse time than now? What will he do to prevent it?
3. Create a barrier between your main characters that keeps them from communicating at a critical moment. Suppose one is telepathic, and the other can't shield thoughts. If something suddenly hampered that flow of thought, what would happen to their usual smooth dynamic? Would the one with telepathy think the other was hiding something? Would this cause a misunderstanding that might fuel conflict?
4. Give your hero/ine an idiosyncrasy. For example, does your heroine tap her chin when she's deep in thought? Perhaps this is how the hero realizes she's hiding something later in the story, and clues him in to make a critical decision. Make the odd habit enough of a part of the character for readers to recognize it too.
5. Create a list of things that your main character wants. Write down three conflicts for each one, and then figure out an unexpected solution. For example: WANT #1: to be a mom more than anything in the world. CONFLICT #1: She can't conceive because _____. CONFLICT #2: Because _____ happened in her past she doesn't trust men. CONFLICT #3: Because she works as a ________ she travels all over the world, so how can she even think about adopting? She's pretty much given up her dream, but it's still there, taunting her. UNEXPECTED SOLUTION: Her work puts her in a war torn area where an orphaned street kid's savvy enables him to save her life. How might saving this child lead to the loss of other things in her life? Career opportunities? Travel? Freedom of movement? How can she reconcile her desires against the child's very real needs, and the fact that she owes him her life? Now, throw in a hero who risks his life to help them both. How might this change her conflicts?
6. What event from your hero and/or heroine's lives changed them the most? How did it come about? What would they do if they could change it? How can you adapt that concept to the current story?
7. Don't be too kind. Let your favorite character suffer. Conflict is the lifeblood of a story. Readers want to see a happy ending that is well earned, so don't go easy on your character. Help them overcome it.
8. Don't be afraid to make your villain truly evil. No one fears a wimp. It's okay to be bad (at least in this case).
9. Consider what your main characters don't want. Is there something they would never do even under penalty of death? What is it, and why?
10. A character was forced into a compromising situation in which a lie seemed the best way out. Over the years, that lie became second nature, and almost seems like the truth. Now, the lie has been exposed. What will happen if the truth comes out? How does your character react? Will he/she try to cover it up? To what extent is he/she willing to go? What emotions go through the character's heart?
11. Create doubt for your character. Yes, we know that the good guy will win, and the bad guy will be defeated, at least for now. But at what point does your hero/ine look at a situation and think "How can I keep doing this? How can I go on?" No action movie is complete without a final showdown in which the hero/ine is so beat up and so close to defeat that you start to hold your breath and wonder "will he make it?"
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12. Foreshadow conflict for your characters. In the book Surrender Love, the two heroes are generations apart. When a cousin points out the large age difference, the younger hero says, "The age of someone isn't something I think about." Later, it's discovered the younger hero isn't as old as he lets others think he is, and the older one isn't just older -- he's immortal. Each has a reason to hide his age, and part of that reason is a character flaw as well as a strength.
13. Make a list of ways your characters handle conflict, and stick to them -- until there's a reason to change. Does your heroine head for the bar, or chocolate? Does your hero stalk off in a huff, or go practice yoga? Does they both go to the shooting range and blow stuff up? At what point does your character talk to a friend and hash out the details? Is he too much of a loner to talk about emotional things? How might he handle conflict if there is no one to talk to but he is a talker? Does he pace? Does she keep a journal? What if that journal is found by the villain, who uses it against the hero? At what point does your character's usual handling of conflict backfire or fail to work? That can be the crux of a conflict all on its own.
Conflict and characterization go hand-in-hand. How your characters react to conflict is what makes them who they are. Write them with depth by knowing how they deal with life. Ask them questions, be willing to listen, and then use those reactions to beef up the story.