The first step in finding a plot for a character based novel is knowing your protagonist well enough to understand how she will make a transformation throughout the story.
Once you feel close to the protagonist and understand her, you can look at how she will drive the plot. It's important to remember that character based fiction does not require a formula—you don't need a 3-act formula, Freytag's 5 Act structure, an 8-point plot, or whatever else you can use in plot based fiction.
What you do need to keep in mind is the character's transformation and your readers' needs.
What do your readers need?
Readers, consciously or unconsciously, want several things from your protagonist when engaged in a character based story. These items involve the transformation of your main character and intrinsically create plot.
1. Readers want to know what motivates the character. Your protagonist needs a clear central motivation. She might have more than one motivation, but there should be one that the others branch from or contribute to. A protagonist could be insecure of his place in his new school and be jealous of his friend, as in A Separate Peace.
The motivation should manifest itself into a goal. The goal might change over the course of the novel, but the motivation will not. For instance, in A Separate Peace, insecurity drives the main character to understand why he would try to hurt his friend, and to find some sort of peace with who he is—his goal.
2. They want to see your protagonist screw up. They may like him, but they like him more when he's human. This humanity is what puts the plot into motion. He needs to do something that causes problems. The protagonist might act on his jealously—he might shake a branch and make his friend fall from the tree. We've all experienced jealously and have acted on it—or we thought about it. Seeing him do this allows us, the readers, to relate to him a deeper level than just liking a character because he is charming.
3. They need your character to stay on unsure footing throughout the story. In the case of A Separate Peace, when the friend falls, he is critically wounded and cannot continue life as he had known it, and he cannot continue with the plans he had for his future. This puts the protagonist on unsure footing because he doesn't understand why he acted on this jealousy. He's generally a good guy.
It's important that he doesn't understand; this builds his transformation into the story so that it doesn't feel tacked on in the end. It also allows the reader to understand a different perspective, a new process in dealing with the things we all deal with. Why doesn't he understand? There are always many different possibilities—this is where your story can take on all sorts of distinctive details.
4. Readers need your character to get it—but fail anyway. He can't be an idiot. Readers want him to figure out what his problem is, but just understanding what the problem is can't be the solution on its own. It's never that easy. Now they can root for him, though.
They need problems to become inevitable, beyond the character's control. He's figured out what his problem is anyway—he's not sure how to act, but he's working on it so he's not totally shocked by what's happening.
5. They want your character to succeed in the end. I mean, you may want him to die to be able to succeed, but that's up to you. At least he succeeds. Continuing to fail is depressing.
Take your time.
You can't rush writing. There is no required number of pages to get from 1 to 2, or 3 to 4, or whatever. You don't have those constraints with a character driven novel. Sometimes 1 is stated, rather than revealed, and the story begins with 2. You have the freedom to tell your story the way you want.
Even knowing these 5 items, you'll find out new things about your protagonist and may need to change what you thought the plot was. It's necessary to be open to this. As Patricia Wrede stated in her blog:
“Tossing the plot outline is hard to make yourself do, but trust me, you’ll be much happier with the result than you will with a bunch of cardboard puppets.”
No one said writing was easy.