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I should patch this one up before too many questions are asked, because I actually do have a method to this madness (the TL;DR is at the bottom of the post):

Bruce Wayne was assuming the identity of Batman for 10 years, watching over Gotham and gradually bringing the crime rate of the city down to a record low.  With many of his Rogue’s Gallery spending their days in Blackgate with life sentences, small underground rings that have been maintained by the likes of Harley Quinn have been keeping the criminal sub-world alive during the midnight hours.  Harley Quinn, who was building her own syndicate with the likes of Poison Ivy, a boy off the street who was exposed to photosynthetic chemicals at birth, giving him is symboitic bond with plant life, moved quickly to recruit stray criminals like Phillis Patencia, codenamed Catwoman. Phillis often crossed paths with Pam Isley while Harley Quinn was creating this new criminal underworld, with Catwoman and Poison Ivy being common suspects in large museum heists and standard bank robberies. 

During this time, Phillis, (then Catwoman) was crafting her own arm of Quinn’s syndicate with a lover of her’s at the time. Her partner in crime, Michael Irons (a high profile cage fighter) would end up being the one and only man Bruce Wayan killed.  During an intense footchase, Bruce aimed to throw a Batarang at the loot bag Irons was holding, but miscalculates and slashes the veins in Irons’s wrists.  Mike Irons bleeds out rather quickly and dies abruptly.  Sure enough, Phillis discovers the body of her boyfriend in the alleyways of Gotham, days after the incident.  

Months pass by, as Batman seems to stalk the streets less and less… To avenge her lover, and revitalize the cashflow of the underground, Phillis crafts a vague plan to find out more regrading Batman. Phillis loads a revolver and broke into Wayan Manor in hopes of looting Bruce Wayan’s house and juicing the millionaire for leads on the whereabouts of Batman. Upon arrival, the door is open, the windows as well.  The Bruce’s study rooms are a pigsty with computers smashed, book shelves were torn down, left in a destructive heap; Wayan was nowhere to be found.  The only clue to Bruce’s whereabouts was Alfred, who Phillis holds at gunpoint until he reveals that Bruce Wayan was in fact, Batman and that Bruce left Gotham at dawn, never living down the guilt of the death of Irons.  Phillis eventually discovers the Bat Cave with the assistance of Alfred and a bit of snooping around the estate.  Downstairs in the cave, equipment being left untouched for months seemingly; clearly, Bruce weighed abandoning the cowl for a long time before his untimely disappearance.  

Phillis has a defining moment here; she empathized with Bruce’s severe guilt, rationalizing his feelings of shame and disappoint after the Irons incident. As crime continued to rise, Phillis took it upon herself to continue the legacy of Batman.  Hoping to instill the fear of petty criminals, Phillis takes the alias of Batman as a fear tactic amongst criminals, (with Batwoman not existing in this universe).  Once she made her decision to take over the cowl, she made a conscious decision to experiment with muscle enhancers and beta tested hormones to pack the same physical strength as a 200+lbs Bruce Wayan.  Unfortunately, Patencia was not prepared for the side effects of the cocktail of chemicals and drugs which would lead to a permanent stunting of her genitalia. 

Ignoring these drastic and dangerous changes, Phillis proceeded with resuming Bruce’s vigilante justice, usually being able to work fast and efficient enough for criminals to not be able to differentiate the old Batman from the new one, except on one critical occasion; against Pam. Isley managed to keep Phillis in a place long enough to discover that this Batman is far different from the one of last winter.  Phillis tries to mask her voice as much as possible especially around Isley, fearing the Pam might find out her true identity.  Phillis has a long friendship with Pam during their days in the underground, see him as her quirky younger brother, taking him under her wing whenever Harley was too busy building the army.  

And the grand finale:

Alfred is helping Phillis out of companionship.  With Brue leaving so quickly and quietly, Alfred stayed in the manor originally out of fear that Pam would kill him as well; essentially being taken hostage.  Later on, when Phillis makes her crucial decision, Alfred begins to see Phillis change into the more collected and calculated individual that her former nemesis once was.  In order to keep up the appearances of the Bat Family, Patencia enlisted the aid of the Watson Twins in order to help in her crusade against the criminal underground of Gotham City. 

Indeed, Catwoman found new purpose in becoming Batman.

TLDR:

• Bruce left ‘cause he killed a guy.
• Phillis uses Batman instead of Batwoman in order to scare criminals (compounded with Batwoman not existing in this universe).
• Drugs.
• Her and Pam were on the same team and she’s afraid Pam knows who she is.
• Because Alfred’s lonely and Phillis stopped being a jerk.