The frog shows his bully personality in the way he responds to the nightingale and how he shapes their relationship.
After "the gaping frog" observed the reaction by the other animals to the nightingale's song, he realized that he had to do something to quiet this rival or he would never be able to croak in that part of Bingle Bog again. The frog was wise enough, however, to understand that a direct assault would not be tolerated by the others.
So, the crafty frog introduced himself as another experienced singer and as "a critic of such note" who appreciated the need for proper technique and training if one wishes to become a truly accomplished singer. The nightingale is thrilled to think that such a respected teacher would be willing to provide vocal coaching, and is eager to do her best to follow all directions under any circumstances.
The frog exploits this willingness to be a completely obediant student by driving the nightingale to sing under conditions guaranteed to ruin her singing voice - cold, wet, without proper rest, extending the range and volume beyond what the nightingale's voice could produce. Finally, the strain becomes more than the nightingale's system can stand; she "tried, puffed up, burst a vein, and died."