oops, just realized you asked about character breakfast, not brunch. Character breakfast was a real blast for our kids. We got tons of great pictures. You will be told what day it is on by your waiters in the dining room (it is on a sea day and they have assigned days and location--I believe it is always in Parrot Cay).
Most all of the characters make it to your table for "photo-ops" so be ready with your own camera as well. You can also have your kids bring their autograph books if they have them--I saw a few characters signing these--wish we had known to bring ours!!!
This is an "order off the menu" sit-down breakfast and is scheduled at a certain time--can't recall what time we had it but since we had early dinner time, we also had the early breakfast time--maybe 8:00 or 8:30? I'm sure others will post more for you--good luck and HTH.