Each day there was a list of characters.. There were lots we didn't see but I know they had:
All the Princesses, Fab 5, Woody, Jessie, chip & dale, Jack Skillington (probably only on the Halloween cruises), Disney Junior characters, Stitch, Tinker Bell, Captain Hook & Mr Smee, Peter Pan, Jack Sparrow.
On pirate night the characters wore pirate gear. On our Halloween cruise they also wore Halloween outfits. You can also pre-book character experiences with the princesses and Frozen sisters.
The only character breakfast we had was Disney Junior with Mickey, doc Mcstuffins, Jake and Sophia. You can also prebook this when your activities open. The breakfast on the western was on the first sea day and grand Cayman day. I'm assuming that on eastern they will offer it first 2 sea days.