One way is through its interactions with different races. For example, if it has had many encounters with elves, it might develop a back story where it learned valuable magical knowledge from them. This could be incorporated into its training and development, making it a dragon with a special connection to elven magic.
The environment in which the dragon is trained can play a huge role. If it was trained in a volcanic region, it could have a back story of having to overcome the challenges of the harsh environment. Maybe it had to learn to withstand extreme heat while being trained, and this experience shaped its personality and abilities. It could be more resistant to fire - based attacks and have a fierce, tenacious nature as a result of its training in such a difficult place.
The trainer or trainers also influence the dragon's back story. If the trainer was a cruel and power - hungry individual, the dragon's back story might involve a struggle for freedom. It could have been forced to perform dangerous and unethical tasks during its training. Eventually, it might break free from its trainer and have a new chapter in its back story where it tries to find a place where it can be free and use its training for good instead of being exploited.