He wound likely have wound up in an orphanage, but still angry and bent on revenge. He would have been a problematic child, always fighting and in trouble. At some point, likely as an older teenager, he would have run away from the home, and wound up in the streets, picking fights with criminals and losing a lot of the time. If this were realistic, he would have gotten shot and that would be the end of the Wayne family.
Staying within the comicbook-type story, he would have met some sort of mentor who told him that picking random fights was not the way to avenge his parents and told him to get training and focus and all that stuff. He would have gone travelling, hiding aboard ships and such. He would have trained with ninja-monks in remote mountain temples, because they don't care about how rich he is (or isn't).
And then he would have gone back to Gotham and been a crime fighter. He'd build himself a batcave somewhere under the city, with cobbled-together equipment. Maybe he'D have some contacts in the police, who give him information in exchange for help with catching criminals because he can be outside the law.
He'd face very different challenges than wealthy Bruce Wayne. He wouldn't have the high-tech gear, but he would be a total unknown. After disappearing from the orphanage, everybody would assume that he died somewhere in the streets, and given the general level of competence in Gothams' police, nobody would wonder too much that no body was ever found. He wouldn't have to deal with all the trouble of maintaining a double life.