I'd call it Shadow of the Batman.
John Blake left the Gotham Police Department and found the Batcave, but he doesn't intent to replace Bruce Wayne as Batman. He has become friends with James Gordon, who has been replaced as the Commissioner of Police and is under federal investigation for covering up the crimes commited by Harvey Dent 10 years prior. Gordon's daughter, Barbara, has moved back to Gotham to attend Gotham University and support her father, and becomes romantically attached to Blake in the process.
Oswald Copplebot, the CEO of Cobblepot Developments, becomes the new Mayor of Gotham and overrules the Dent Act, leading to thousands of legal processes to short out the arrests that Gordon has made under it. Because of this, the streets are filled with criminals once more, and a new mobster, Tony Zucco, is reuniting them under his rule, creating a crime syndicate that not even Carmine Falcone and Sal Maroni were able to in their time.
Zucco is supported by mercenaries David Cain and Floyd Lawton, and Blake soon learns of his existance. He and Barbara begin investigating the matter without police protection, and end up discovering that Zucco is being supported by Cobblepot, the real mastermind behind the organized crime.
Blake tells Barbara the truth about Batman, and decides to take up the cowl in order to stop Cobblepot before more innocent lives are lost. Blake trains, and gain some knowledge of combat, though he is still not a match for Wayne. Barbara helps him perfect Bruce's equipment to suit him, and Blake begins sabotaging Cobblepot's operations and gathering evidence of his crimes to present to the police.
The movie develops Blake's confrontation with Cobblepot, him easying into the role of Batman, and his final acceptance of the mantle and the rise of Gotham's new protector. Intercut, is the birth of Blake's nemesis, FBI profiler Edward Nygma, who assists in the investigation of Gordon and slowly becomes obssesed with the new Batman and descends further into his need to "solve the riddle", becoming the Riddler in the fifth movie.