The character meals at WDW are structured so that the characters make it around to every table in the restaurant approximately once an hour. You can plan on this being quite close to accurate at a large character dining venue such as Chef Mickey's. At a smaller venue, such as the Cape May Cafe, you may get the full character rotation in significantly less time than that. So, you can plan on an hour for nearly all character meals. If you're at a non-buffet character meal, such as the Garden Grill or Cinderella's Royal Table, you could be a bit longer if you were going to linger over coffee and dessert. Or if you're at a buffet and you're in a hurry, you could probably be done at most places in 45 minutes or so. You can read the answer to a similar question
here.
As for travel time from the Contemporary to the parks ...
You could get from the Contemporary to the front gates of the Magic Kingdom in about 10-15 minutes walking, depending on your pace. Or in about 15-20 minutes using the monorail depending on whether you have to wait for a train to arrive.
You would get to the other parks by hopping on the monorail and getting off one stop later at the Transportation and Ticket Center. From there you would take the Epcot monorail line to get to Epcot or a bus to get to either the Animal Kingdom or Disney's Hollywood Studios. The actual travel time to the TTC is about 5 minutes, but I'd leave a 5-10 minute buffer there in case you have to wait for a train. The travel time to the other parks from the TTC is in the 20 minute range. But there again, I'd leave a 10-15 minute buffer for transportation waits. Additionally, you'll have to add in a few minutes to go through the park security lines and entrance turnstiles and to walk to your restaurant. All-in I'd leave at least an hour in advance for any character meal at Epcot, DHS, or the AK.
Hope this helps.