Il s'agit de Harley Quinn (fictif)

Most natives would tend to consider the <h> of "Harley" as non aspiré and so say "d'Harley Quinn", but both versions would be right

Harley est une personnage fictif de DC.

"personnage" is always masculine, regardless of who the character is

Elle a travaillé comme une psychaiatre dans à l'asile d'Arkham asile,et elle est tombée amoureuse du Joker.

You don't need articles when simply giving a title or profession: travailler comme psychiatre, avocat, maçon, agriculteur, chanteur, être enfant/adulte, être roi, prince etc

"DANS l'asile" would emphasize "within, inside the building"

French has an inverted logic compared to English: the main word comes first (here, "asile") and then what completes it. Here a "de" would be required to connect both, lit. "the asylum of Arkham"

Elle, s'est finalement, se fait jetlarguer et a été mise à l'asile comme son petit-ami.

"se fit" is passé simple. You've started with the passé composé though, so you have to keep on with that tense
"se faire larguer" is way more common in the specific case of a bf/gf dumping his/her partner
"finalement" as parenthetical element between "elle" and the verb wouldn't be used

Elle s'est enfermée dans l'asileà l'intérieur.

Past tense => s'est enfermée
"à l'intérieur" ("inside") to avoid repeating "asile" too much

Harley se'est fait jeter dans un acide que coloreri a rendu sa peau blanche comme le Joker/comme celle du Joker.

l'acide (subject) a rendu = "qui", subject relative pronoun => un acide QUI...
"colorer EN BLANC" would be the right way to put it, but it would mostly work for something getting dyed or painted, not destroyed by acide