Bruce Wayne, Jr. was the biological son of Bruce Wayne of an alternate reality in which the elder married and had children. Bruce Wayne Jr. did not truly appreciate his father's legacy, although he did train to the same levels both physically and mentally (as did Dick Grayson, who became Wayne, Sr.'s junior partner Robin).
Unlike Grayson, who was appreciative of the Batman affiliation, Wayne Jr. resisted his father's heroic legacy; he had not gone through the same personal loss of family as his father and Grayson. Wayne, Jr. also did not want to be known merely as the son of his famous father. Because of their similar situations, being thought of as mere successors to famous fathers, Wayne Jr. and his friend Superman, Jr. went on a road trip in a period of self discovery. It is during this excursion that Wayne, Jr. determines his own sense of sense and decides to become the successor of his father in his own way. Superman, Jr. has come to a similar realization and together they form the team known as The Super-Sons.
When Bruce Wayne Sr. was murdered, Wayne, Jr. reluctantly teamed up with his father's former ward Grayson who was more appreciative of the Batman identity, even challenging Wayne, Jr. for the right to become the next Batman. Grayson and Wayne, Jr. put aside their differences and successfully teamed up to find Wayne, Sr.'s killer who was revealed not to have actually succeed in killing Wayne, Sr. Grayson and Wayne, Jr. agreed to disagree who would be the next Batman though came to respect each other's abilities and determination as both successful protegees of Batman, Sr. in their own respective identities.
Bruce Wayne, Jr. along with all members of his reality were erased from reality during the Crisis on Infinite Earths
, which retroactively erased him and his reality from ever having existed.