The plane was attempting to land, but pulled back up at the end of the runway.
"It attempted to abort the landing and take off again," Sgt. F.L. Tyler of the Virginia State Police told reporters. "It then went off the south end of the runway and then landed in the tree line."
When the plane struck the trees, it immediately caught fire, officials said.
Rescue workers recovered all six bodies from the burned wreckage.
Virginia State Police identified the victims as:
• William C. Hamerstadt, 64, of Carmel, Ind. (pilot)
• Robert D. Ross, 73, of Louisville, Ky. (owner of the plane)
• Lisa K. Borinstein, 52, of Shelbyville, Ind.
• Luke J. Borinstein, 19, of Shelbyville, Ind.
• Emma R. Borinstein, 15, of Shelbyville, Ind.
• Maren Timmermann, 15, of Berlin, Germany
The aircraft departed Louisville on Friday morning, stopped in Shelbyville and continued to Fredericksburg.
The cause of the crash is under investigation. Officials with the National Transportation Safety Board, Federal Aviation Administration and rescue and fire crews were at the scene.