I have never heard of having a day with a character in the parks. I feel confident that this would not be possible because it would be so frustrating for the other guests in the park at the time. Characters can be booked for special events such as weddings; this costs many hundreds of dollars per half hour session. I don't even want to think what the fee for a day would be.
The characters do, however, sometimes give individual children individual attention for brief periods of time. For example, Goofy often rides on the Barnstormer with a child first thing in the morning. My oldest daughter was once pulled out of the crowd by Minnie Mouse and taken to ride on Dumbo with her. This was obviously a highlight of that trip and we have photos we'll cherish forever, but there is no way to arrange for this type of experience in advance. You just have to have a bit of luck and be randomly selected. For other concentrated character time, you could always dine at one of the many character meals at WDW, or you might also want to look into the Family Magic Tour at the Magic Kingdom. You can read a bit about my experience with this tour
here.
As far as large groups of characters in one place ... Toontown is a great place to do this. There is also a similar character greeting zone in Future World at Epcot. I have often seen groups of 5-10 characters greeting guests at once near the train station at the Magic Kingdom and near the Sorcerer's Hat at Disney's Hollywood Studios. But these situations come and go depending on the weather, crowd levels, and many other factors. If you want to see as many characters as possible, you're going to have to travel around the World a bit.