There is an organized event that occurs on a few mornings throughout the cruise. Children are free, and adults are $10. It's the same menu that you can get in the regular Main Dining Room, but renamed for the characters.
There are typically three characters that come out, and it varies with each breakfast. We had Po and Tigress from Kung Fu Panda (something my son doesn't watch) and Puss in Boots (who we had already met earlier in the week). So it wasn't as exciting for our son as we had hoped! It was still worth it, and the kids in attendance had a lot of fun!
A nice improvement, since the last time we did a character breakfast in 2012, is that they give the kids a sticker when it's their turn to meet the characters. This helped control the flow of people, and no one rushed up to them!