The film doesn't clearly explain Max Rockatansky's origin other than brief flashes of his presumed daughter being killed. However, in the original Mad Max he was the top driver of the Main Force Patrol (aka 'The Bronze', a derogatory moniker used by the biker gangs Max tangles with), a national police force attempting to preserve order on Australia's highways as civilization begins to crumble. The film showed Max's family being run down on the road by a biker gang which led to him seeking revenge on those who murdered them, then heading out into the wastelands (the sequel suggested a nuclear war followed shortly afterwards, destroying an already crumbling society). A featurette on the character can be viewed here. Max also wears a near-identical outfit as he did at the end of Mad Max and the entirety of Mad Max 2: The Road Warrior. The right sleeve is cut off his jacket because a motorcycle ran over his arm and paramedics had to cut the sleeve off. He carries tools on him at all times for running repairs on his car. He drives the same modified police Interceptor. He also wears a leg brace due to being shot in the knee.
There is a fan theory that Max in this film is actually the Feral Kid from The Road Warrior and that he simply modeled his outfit and car from his idol, Max Rockatansky. The evidence in support of this is: When we first meet Max in this film, he has a very unkempt look (more so than Mel Gibson's Max ever did) with long shaggy hair and a beard. Throughout the film, Max says very little, often resorting to grunting or grumbling (which is all the Feral Kid did in Road Warrior until it was revealed at the end that he was the narrator, now an old man). Max's flashbacks in this film reveal he had a daughter who was run over by a gang but we don't see his wife at any point. The original Max had a wife and son who were run over by a motorcycle gang. We see several other people appear before Max in this film, people he failed to protect. These may have been surviving members of the group from the Road Warrior, of which the narrator mentions he became the leader. So it's possible the Feral Kid grew up, became the leader of the group and had a daughter, all of whom were killed, which led him to taking on the mantle of the Road Warrior, living in exile.
The counter-argument to this theory is that George Miller, the creator of the Mad Max universe, has said this character IS Max Rockatansky. The opening narration by the character says "My name is Max. My world is fire and blood. Once, I was a cop. A road warrior searching for a righteous cause. As the world fell, each of us in our own ways were broken. It was hard to tell who was more crazy... me... or everyone else."
This implies he is Max and has Max's background of being a police officer before the nuclear war. However, he could be telling this story to whomever is listening in order to pass off Max's background as his own. At the end of the film, he also tells Furiosa that his name is Max, though he says it in such a way that suggests that he may be trying to convince himself of it as much as her, or perhaps that he hasn't used it for so long that it seems strange to him.