(Arkham therapy scene)
"Dr. Quinzel, I live for these moments with you, what do ya got?"
"So thoughtful. There's something you could do for me, Doctor."
"I need a machine gun."
Insert huge metallic grin
--------------------–------------------------------------------------------------ (Arkham raid/Harley Torture scene)
Enters Arkham room, waving arms
"What do we have here?"
transition to him standing over Harley
"What? Oh, I'm not gonna kill ya. I'm just gonna hurt ya really, really bad."
------------–------------------------------------------------------------(Monster-T club scene)
"Are ya sweet talkin' me?
"Ha-ha-ha. I love this guy, he's sooooo intense!"
"Oh that she is. The fire in my loins, the itch in my crotch! The one, the only, the infamous Harley Quinn!"
"Oooh, come to daddy."
"Listen, you are my gift to this handsome HUNKA HUNKA! You belong to him now!"
"You don't want no beef? You don't want no beef? You don't want no beef?"
"Are you enjoying yourself?"
"That's right!"
--------------------–------------------------------------------------------------(Batman chase scene)
"Ooh, we've got company!"
--------------------–------------------------------------------------------------- (Finding Harley/Weapons scene) "Where is she?"
"Bring the car around, we're going for a drive."
-------------------------------------------------------------(Grigg's intimidation scene)
"Blah blahblah blahblah blah blah. All that chit-chat's gonna getcha hurt."
"I can tell you meant that!"
Sits on Grigg's lap, strokes his face
"You're gonna be my friend."
--------------------–----------------------------------------------------------(Nanotech raid scene)
Grabs nanotech gun
"Ooh, this looks neat!"
Stabs guy with said nanotech gun
------------–----------------------------------------------------------(Chemical wedding scene)
"Question. Would you die for me?"
"That's too easy."
"Would you- would you live for me? Hmm?"
Covers Harley's mouth with his smile tattoo
"Careful...DO not say this oath thoughtlessly! Desire becomes surrender, surrender becomes ..POWER."
"Do you want this?"
"Say it...say it. Saaay it. Prettyprettyprettyprettyprettypretty..-"
"Mmm, God you're so... good!"
--------------------–---------------------------------------------(Helicopter/rooftop scene)
Speaking over the intercom to the professor that is disarming Harley's nanobomb
"Professooor, could you PICK UP THE PACE?!"
"Come on baby.."
"Oh you know I'd do anything for you.."
"By the way, I've got some grape soda on ice and a bearskin rug waitin'."
"Huh? "
"This bird is baked. Okay honey, it's me and you!"
--------------------–--------------------------------------------------------------- (Ending breakout scene)
"Let's go home."
-Joker, Suicide Squad
--------------------–--------------------------------------------------------------
The Therapy scene showcased their initial relationship.
The Torture scene showcased how everything changed, including Harley herself.
The Club showcased the Joker's status but more importantly, his public relationship with Harley.
The Chase scene elaborated on their separation and Harley's imprisonment.
The Weapons scene was him tracking her and discovering the presence of Argus. (Not specifically)
The interrogation scene was showcased the Joker at work, flexing his status and intimidating someone simply by being himself. Being manipulative, being him.
Nanotech Raid scene showed off his resourcefulness, tracing the bomb back to the company and confronting the scientist(Who's wife he already had in jeapordy) that would eventually deactivate Harley's bomb. Guns blazing, chaos afoot, much like the Arkham raid.
The Chemical Wedding scene solidified their relationship, the final step in the Joker's seduction, the completion of her origin. The silver tongued Joker is a lethal one after all, taking advantage of her dependency.
The Helicopter scene mainly showcased the interest between the two after a long time of separation, the damaging of a naturally chaotic relationship as the Joker seemingly dies.
The Final scene was pure greatness and gave Harley a reason to be out and about.
The purpose for dissecting those scenes was to elaborate on how the Joker's main purpose in the film was to perpetuate Harley's growth as a character, as well as their relationship. Yes, his personality is also highlighted multiple times but this is why he made the most of the time he was apparently given. I can say I am excited for a film that focuses on this Joker as a whole.
EDIT: Some context, fixed quotes, added scene interpretations.