Make a story that focuses on the iconic four, joker, penguin, catwoman, and riddler, make each of them on their own a formidable threat to batman by making each of them comparable to batman in some specific area.
joker is the exact opposite to batman in every way, but also an equal, the guy has three personalities in one head, and all three have their own personality, capabilities, and origin story.
One is a jack nicholson golden age type, one is a ceasar romero silver age type, and one is the heath ledger bronze age type, the last is one that combines all three into one unstable amalgamation that averages the three out, and this one is aware of the fourth wall of fiction and reality, his origin is that he is a character created for batman to fight by a team of artists, but this personality is rarely in control for long, often only appearing in the transition between two of the other three.
penguin is another story, his relation to batman is that he has more wealth, power, and a better reputation than bruce wayne, he was born to a wealthy family who had been one of the founding families of gotham, whie normally this would ensure a life of ease, he was not so lucky.
he was born twisted and deformed as a result of coming from incest, his problematic birth also meant his mother could not bear another child, and his father had trouble with his ability to sire a child, thus he was abandoned and replaced with a boy kidnapped off the street, winding up an a circus, spending his childhood as a performer in a freak show.
he eventually made his way back home, but by then the changeling had squandered the family fortune, so the penguin secretly killed his replacement, collected the sizable fortune off of his late brother's corpse, and established himself as a legitimate businessman in the vein of lex luthor, another villainous self-made man.
he now has his own megacorporation that has become larger than waynecorp, but this isnt just due to a good business acumen, the truth being that he also took advantage of the decline in organized crime due to some events at the time in order to position himself as the new leader of the remains of their criminal underworld.
he is now known as a billionaire philanthropist and a gentleman who has not only shot past even bruce wayne in his fortunes, but also in the public favor, if only they knew the real secret of his success and what he is really doing with his power, which is being the power behind the money of pretty much the entire criminal world in gotham, a "face" who sets things up, launders the returns, smooths things over, and takes his cut, his only concerns are improving his wealth and reputation.
the riddler compares to batman by being smarter than him, the moriarty to batman's sherlock, he was a genius from birth, one who was plagued by boredom for most of his life, as an adult, this would manifest as a part of the reason for why he would always give people a way to beat him, a way to escape his traps, a way to arrest him for his crimes, a way to stop his plans, he always does this just because it allows him to have some fun, but also a way to mock those who challenge his massive intellect.
he was entered in an infamous mathematics contest that was designed to be unwinnable, so when he won it at a young age, everyone was blown away, the truth was that he had found a way to cheat the contest so that he would win.
he was gifted at everything he set his mind to, but he was consumed by one passion, which was that he was obssesed with knowledge, it is for this reason that his criminal aims as an adult are to know things, he wants information, secrets, and the like, his first foray into crime was as a detective, one for whom any crime was easily solvable, and the largest obstacle to closing any case was the detective's own reluctance to just explain everything, and instead try to lead on his fellow investigators with subtle clues as to how to answer the riddle in front of them, it was one such riddle that would later be his start of darkness.
riddler had figured out batman's real identity as bruce wayne shortly after discovering the mystery of who the vigilante's identity was, he never divulged this information to anyone else, but would later use it to great effect against the only man in gotham who can match his mind.
the obviousness of batman's identity and the revelation to him that gotham's police force had indeed not figured it out was actually the first catalyst in a chain of events that turned him from a force of good into one of evil, another being that he had found himself doing double duty as a consulting criminal, helping criminals to carry out their crimes in a better way, which he did out of a need for stimulation rather than profit, often his price was only things like secrets, which he had used to advance his own career, leaving clues as to his involvement for additional thrills, until batman exposed him.
now he acts as the man with the plan, coming up with ways to achieve the impossible for the low price of some secret knowledge, the only thing he requires is to leave his subtle calling card at the scene of the crime, a clue as to his involvement (which his clients are aware of) and his client's identity (which they are most often not aware of), he treats his crimes like a game or puzzle, since games and puzzles are something he is obsessed with, he often uses these things as his theme.
when he kills, he places his victim in a situation that will kill them unless they find a way out of it, which he refers to as his "trap games", but he always includes a way out, he is a social darwinist, and those who die deserve to do so in his eyes due to how stupid they were for failing to see what to him wold have been an obvious solution to their problem.
for the riddler, batman is an endless source of amusement and a challenging mind to set his own against.
catwoman is someone who is doing wrong things for the right reasons, a former office worker by day and prostitute dominatrix by night, she was left with a responsibility to care for her schizophrenic brother and the many young children who lived on the streets outside her home.
she found opportunities for this in robbing her clients during their visits in order to care for the children, and later she graduated to breaking into their homes, some of her clients were wealthy and esteemed members of her city's upper crust, which gave her an opportunity to discover her talent for navigating past high levels of security, soon she was a thief for hire, if you wanted something from a museum, she would be able to get in and out with it, leaving not a trace behind.
she was a catburgler whose outfit resembled that of a cat, and her brother called her "alleycat" as a nickname due to her childhood love of milk, thus she became known as the catwoman, she's not a bad person at heart, and genuinely wants to make the city a better place for the most vulnerable members of her society, but this had led to her becoming more and more violent as time went on and things got worse, cynicism and an encounter with the league of shadows and seen her graduate from a thief to an assassin, targeting the most evil people in her city who couldn't be touched by the law.
the joker, penguin, and riddler, are now giving her an offer that could see a real change in how things operate in her city, a way for things to get better, she's willing to side with what she sees as a lesser bottom-level evil in order to take down a larger evil at the top of the system, little does she know that she's a tool to them, to be disposed of when she's no longer of use (she's too naive and good intentioned).
she's not along, both penguin and riddler are going to get treated the same way by the joker in his plans, even though they are planning to betray him (because he's too dangerous and unpredictable).
joker is batmans opposite
penguin is batman's for