A lot of spring breaks are taking place April 7-11, so that will be the busiest week in April. Also, the Rod Run in Pigeon Forge is the 10th-12th, so that week is going to be extra-packed in town.
The 2 weeks after that will be much lighter.
You can check the Dollywood Crowd Calendar to get an idea of the best/worst days to visit. Looks like April 21, 23, or 25th would be your best days in April.