The plaza is the only character dining in DL, and yes, it is only for breakfast. I find that character dining isn't really necessary here. Because of the intimate nature of the parks, I have never had a hard time finding characters (when I wanted to), except perhaps around September, which is always the most empty time of the year, but allows for plenty of quality time in toontown.
Plaza Price
Breakfast - $17.99 Adult/$9.99 3 - 11/no charge under 3
As per everywhere else:
DCA:
Avalon - Character dining at Lunch and Dinner ( I am unsure about Breakfast). It is an order off of the menu place, but I have had a more relaxing, yet character interacted meal here than anywhere else. I loved the food, and have always had a hard time finishing it. Kids meals are served in a sandpail with condiments in the shovel. (I think they do this at a resort in WDW, but I don't remember which one - maybe Cape May). Has many options from Pastas to Pot Pies to Seafood. Older reviews were very poor, (when it first converted from Wolfgang Pucks to Character dining), but since early June, every time I've gone was fantastic. The penne pasta is my fave. Mmm.
For the Lunch Price, you get a starter (soup or salad) an Entree and Cotton Candy for dessert. (You can order other deserts for a fee, but it is much less than $5 for cake everywhere else..I think it was $2.50 - The chocolate cake with Carmel Corn extra was very good. For the dinner price (though I have never been) you are supposed to get a starter, Entree, Choice of Yummy desert plus cotton candy. Drinks are extra.
Lunch - $12.95 - Unsure about kids
Dinner - $19.95 - Unsure about kids
Disneyland Hotel:
Served Buffet Style:
Goofy's Kitchen - Features character buffets for breakfast, lunch, and dinner. The breakfast buffet has a yummy spread including Mickey Mouse waffles, and the lunch and dinner buffets feature carved beef and ham as well as several other entrées, a dessert bar, and salad bar. While you eat, the "classic" characters come around for a visit. Last time I went (a while ago) they had a kids buffet with chicken nuggets (mickey shaped of course) mac' and cheese, and mini corn dogs.
Breakfast - $16.95 Adult/$9.95 3 - 11/$3.95 under 3
Lunch - $17.95/$9.95/$3.95
Dinner - $26.95/$9.95/$3.95
Grand Californian:
Storytellers:
Happy Campers Breakfast with Chip & Dale.
This offers a breakfast buffet, but if your kids are picky eaters, and you're not, or vice versa, you also have the option of picking individually priced a la carte items off of a menu.
Buffet Price
Breakfast - $17.95/$9.95/$3.95
Paradise Pier Hotel:
Minnie & Friends breakfast:
Haven't been to this one. I know that some folks like it and some don't...however, I do know that often they have hard to find characters like Merlin & Gaston.
Breakfast - $16.95 Adult/$9.95 3 - 11/$3.95 under 3
Practically Perfect Tea:
Tea Cakes & Crumpets - Ah, Poppins

Only available on Wednesdays & Weekends this is a really fun experience. There are two teas - Breakfast and afternoon. While breakfast serves crepes, crumpets, cakes, scones and quiches (and of course Mickey waffles), some items change to sandwiches and deserts at noon. This has very limited seating, and is a fun experience, even though it is the most expensive in the DL resort!
$21.95 for adults and $13.50 for anyone under 11. Make an 11:30 ressie, and you may be able to have a taste of both teas.

~Tina
All of these places can be contacted by calling for priority seating at (714) 956-6755.