The personality of a character in a play is revealed primarily by (1) what the playwright says about the character, (2) what other characters say about the character, and (3) what the character says about themselves.
What the playwright and other characters say about a character are considered direct characterization. What the character says about themselves, how they act (alone and with other characters), and how other characters react to them are considered indirect characterization.
In Othello, the playwright, William Shakespeare, says nothing about Desdemona. Shakespeare provides no description of Desdemona in introductory notes to the play or in stage directions within the text of the play itself.
It wasn't until the publication of the Fourth Folio in 1685, nearly seventy years after Shakespeare's death, that Desdemona was described in a list of characters as "Desdemona, wife to Othello." Until then, all published version of Shakespeare's plays contained no direct reference to Desdemona (or to any other character, for that matter) except in the text of the play.
Desdemona isn't mentioned in Othello by name until the second scene. In the first scene, she's described by Iago and Roderigo only in terms of what she's done. She's betrayed her father, Brabantio, and secretly eloped with Othello.
RODERIGO. Your daughter, if you have not given her leave,
I say again, hath made a gross revolt,
Tying her duty, beauty, wit, and fortunes
In an extravagant and wheeling stranger
Of here and everywhere. (1.1.144-148)
The woman who Roderigo and Iago describe would appear to be reckless, impetuous, lustful, and deceitful.
Othello believes otherwise when he first mentions her name.
OTHELLO. For know, Iago,
But that I love the gentle Desdemona. (1.2.26-27)
Brabantio, too, finds it difficult to believe what Iago and Roderigo told him.
BRABANTIO. A maiden never bold;
Of spirit so still and quiet that her motion
Blush'd at herself. (1.3.105-107)
In her first appearance in the play, Desdemona's demeanor is in direct contrast to Iago and Roderigo's description of her. She's quiet, calm, composed, respectful, and extremely well-spoken.
DESDEMONA. My noble father,
I do perceive here a divided duty.
To you I am bound for life and education;
My life and education both do learn me
How to respect you; you are the lord of duty,
I am hitherto your daughter. But here's my husband,
And so much duty as my mother show'd
To you, preferring you before her father,
So much I challenge that I may profess
Due to the Moor, my lord. (1.3.194-203)
Desdemona demonstrates an uncommon independence of mind and spirit in defying her father, and society in general, by marrying Othello.
Desdemona also displays a mischievous, witty, even high-spirited side of her personality in her scene with Emilia and Iago in act 2, scene 1, and Othello must contend with her considerable intelligence and wit in act 3, scene 3, when she intercedes on Cassio's behalf.
Her intercession with Othello to restore Cassio to his position as lieutenant also demonstrates her strong-minded and forthright character and her belief in justice and fairness.
Desdemona is not particularly worldly, however, and she seems naive about marital infidelity.
DESDEMONA. Dost thou in conscience think—tell me, Emilia—
That there be women do abuse their husbands
In such gross kind? ...
I do not think there is any such woman. (4.3.64-66, 88)
Iago uses Desdemona's good and admirable qualities against her to destroy Desdemona and Othello's lives.
Nevertheless, Desdemona never betrays her principles or her devotion and love for Othello. Even though she maintains that she's “guiltless” of infidelity to Othello, she takes responsibility for her death and forgives Othello for killing her.
DESDEMONA. I never did
Offend you in my life ...
A guiltless death I die.
EMILIA. O, who hath done this deed?
DESDEMONA, Nobody; I myself. Farewell;
Commend me to my kind lord. O, farewell! (5.2.68-69, 146-149)
Othello is the Aristotelian "tragic hero" of the play, but Desdemona is no less tragic and heroic in her own right.