There are two possible ways to do this: by phone or online.
To make reservations by phone for any WDW restaurant, including character meals, call the Disney Dining reservations line at 407-WDW-DINE.
The links to make online dining reservation are part of each page on
the WDW website dedicated to individual table service restaurants. For
example, if you wanted to make a reservation for the Chef Mickey character dining experience, go to the WDW
home page at disneyworld.com. In the box that says "Search
DisneyWorld.com" (NOT the box that say "Search Disney.com"), type in
the words "Chef Mickey's", and from there navigate to the dedicated page for Chef Mickey's, found
HERE. You'll see a box on the middle left side of the page that says "Make a
Reservation." Click there and follow the instructions to make a
reservation. To make more reservations, search for the names of other
table service restaurants and character dining locations. It sounds complicated, but it's really quite
easy once you get the hang of it.
Happy dining!