If you want to write about characters without a plot, you need to identify those characters really well. Each character must have a unique trait. This will really help you understand their behavior in certain situation. Having said that, you must essentially have a plot to tell a story.
You can improvise on your story and make it character-driven but you need to have some kind of a plot, be it adventurous journey of a dysfunctional family or a horror, where a haunted family of ghosts are doing some actions. These stories have a thin plot but end of the day, plot is definitely essential.
I have this problem all the time. Normally, I try to think of what would cause the most change in my characters. Character growth is a beautiful thing, but our characters often have to be torn down by bad circumstances in the plot before they can become stronger.
Figure out your character's weaknesses and push them to the limits.
Start writing about the characters. Just every-day life stuff. Then have something unexpected happen, like somebody dies or reveals a terrible secret or gets a new best friend. Write about the character's reaction, and screw with 'normal' events as much as possible. Try to throw in as many plot twists as you can. After the first unexpected event, don't let anything be normal or predictable again. This might help: http://www.rangen.co.uk/writing/plotgen.php
Figure out what these characters want, then figure out how not to give it to them. Put them in extremely conflicting situations and see how they react.
I agree that you have to know your characters well, but when you only have the characters, you have to know how they'd react. This is actually a good thing— it forces you to break away from standard plots that have things happening for no reason; you have to give actions a reason.
I would suggest to pick a fatal flaw for each character and figure out what's going to clash with it, then put your characters in situations like that as much as humanly possible. Then you force them to make really hard choices, and they grow. That helps give your plot richness.
A writer is a world trapped in a person— Victor Hugo
Ink is blood. Paper is bandages. The wounded press books to their heart to know they're not alone.
You don't actually need a plot. I mean, if you know your characters well enough, that is. And it kind of depends on genre and what you hope to accomplish with the work, but I do this thing with realistic fiction where the characters and their lives are the plot. Within a life, there are thousands and thousands of little arcs and things that you learn about yourself and parts of your life that you push though, and, in my opinion, nothing's more fascinating to read about than that.
If you DO need a plot, look at your characters and, as Rosey said, figure out what they want. Next, I create my setting. Once you have your characters, their basic needs, and where/what they have to work with to meet their needs and wants, figuring out a plot is a thousand times easier.
Sometimes, it pays to just start writing and figure out a plot as you go along, too.
