Here's what I've worked out so far:
Hachiman Hikigaya = Batman
"I'm the Hachiman."
Haruno Yukinoshita = Joker
"Why so serious Hikigaya-kun?"
Yukion Yukinoshita = Commissioner Gordon
An exceptional person who disagrees with Batman's methods on principle, but ultimately must admit to their validity and works alongside him
Yui Yuigahama = Barbara Gordon/ Batgirl
She admires both the Commissioner and Batman and tries to be there for both
Saika Totsuka = Robin
The only person who truly looks up to Hachiman and would like to learn from him.
Komachi Hikigaya = Catwoman
She genuinely cares about Batman, though she can cause a lot of trouble for him. Also, cat fangs.
Hiratsuka Shizuka = Alfred
The wise, elderly influence in Hachiman's life who recognizes his potential but tries to guide him away from a path that will destroy him.
Zaimokuza = The Creeper
This one's rather obvious
Saki Kawasaki = Katana
She tends to be quite stoic and determined with anything she sets her mind to.
Hayato Hayama = Harvey Dent
He has a two-faced nature when it comes to dealing with the world's problems. He claims to want what's best for people, but is often unwilling to do what is truly right to meet this end.
Miura Yumiko = Poison Ivy
She's very sure of herself and very cold towards others
Minami Sagami = The Penguin
She's quite petty and has no real power of her own but what she can get others to do for her.
Tamanawa = The Riddler
He's actually very intelligent, but puts his pride on looking intelligent than acting that way.
Isshiki Iroha = Harley Quinn
Always smiling, deceptively cunning, but easily thrown off guard by her love interest.