The OP already hinted at the best way to have fun character interaction. Ask questions about their story.
I've had many fun moments with Belle simply by asking her about what story she's read recently, what her favorite story is, what kind of stories she likes, etc. And don't expect a specific response. I'm not kidding when I say that the following stories have come up in character interactions with Belle over the years: Peter Pan, Star Wars, Les Miserables, A Christmas Carol, and Sleeping Beauty.
You can also try asking about their friends and significant others. Gus and Jaq have lead to two very memorable interactions with Cinderella. I met her at Epcot and asked if she was worried about what they might do in the castle that night. She told me they were baking a cake, so she hoped the kitchen would still be standing when she returned. A few months later in
Disneyland, I gave her the backstory and then asked her what happened. She proceeded to tell me that they had baked the cake out of cotton candy and made a huge mess in the kitchen that took hours to clean up, but that they were her friends and she still loved them.
Also, dress for success. If you have a shirt or something with a favorite character or characters on it, wear that shirt. I've discussed Star Wars, World of Color, Fantasmic, even where I went to college (she had a "friend" who knew my alma mater well) and all sorts of things with Sleeping Beauty.
Above all, give the characters a chance to show off their stuff. When I asked Chip and Dale when Disney was going to come to their senses and give them a movie since they were infinitely cooler than those squirrels (Alvin and the Chipmunks) who somehow got 3 movies, they proceeded to show me their movie skills. You've never seen The Matrix until you've seen it recreated by 5 foot tall chipmunks.