How is Pearl like her mother? How is she like her father?
From direct and indirect characterization, we can infer, as well as deduct, a lot of assumptions regarding Pearl's inherited physical, behavioral, and spiritual traits.
In chapter VI, Pearl is described as physically beautiful, with a "splendor" that comes from her looks as well as from the intensely extravagant clothes that Hester makes for her "with a morbid purpose that may be better understood hereafter".
Just as Hester abides by the code of utmost humility, which most Puritans obey, she also ensures that Pearl drastically and dramatically stands out, like an "exotic bird", as she was described once. This is a dimensional manifestation of Hester's own spiritual state; wanting to explode out in the world, be seen, be known, and be free. She vicariously does all of this through Pearl.
Pearl's physical beauty undoubtedly comes from both her parents, Hester Prynne and Arthur Dimmesdale. Remember that Hester is described in Chapter II as comparably better looking than her counterparts.
The young woman was tall, with a figure of perfect elegance, on a large scale. She had dark and abundant hair, so glossy that it threw off the sunshine with a gleam, and a face which, besides being beautiful from regularity of feature and richness of complexion, had the impressiveness belonging to a marked brow and deep black eyes.
Pearl also inherits looks from her father. In chapter 10, it is explained that, back before the incident with Hester (her pregnancy, that is), Dimmesdale, now sickly and decimated, was also considered once as a very eligible bachelor whom many ladies would have wanted to marry. He was English, smart, sophisticated, charismatic...the "young divine". Pearl had the looks and charisma of her parents.
There is a dimensional difference between Pearl and other children that has to do with the chaotic nature of her conception: of the fact that she was born presumably out of the passion of two people who were not meant to be together.
Pearl's aspect was imbued with a spell of infinite variety... Throughout all, however, there was a trait of passion, a certain depth of hue, which she never lost; and if, in any of her changes, she had grown fainter or paler, she would have ceased to be herself
In chapter 6, Hester correlates Pearl's unruly nature to the state of Hester's own soul on the moment that Pearl was conceived. In other words, that Pearl inherited Hester's passionate spells because she was ill-conceived in a moment of profound passion and emotion- something unheard of by Puritan law regarding sex or romance.
Hester could only account for the child's character[...]-by recalling what she herself had been, during that momentous period while Pearl was imbibing her soul from the spiritual world, and her bodily frame from its material of earth. The mother's impassioned state had been the medium through which were transmitted to the unborn infant the rays of its moral life...
Pearl was a handful, and she seems to also have acquired her rebel nature from both Hester and Dimmesdale. Both Hester and Dimmesdale broke the Puritanical rules. In fact, it is arguable that they would have continued breaking them over and over if Pearl had not happened. So there is a lot of unruly on both ends.
Moreover, there is a streak of perversion that continuously comes up regarding Arthur Dimmesdale which very well may have been inherited by Pearl. Notice how in chapter 10, Chillingworth insists that he sees some wild streak underneath the facade of the "young divine". Something feral and very much human that seems to be pushed down by this "perfect" man.
This man," said he, at one such moment, to himself, "pure as they deem him,--all spiritual as he seems,--hath inherited a strong animal nature from his father or his mother. Let us dig a little farther in the direction of this vein!
From Hester, we know that she is entirely against the status quo of the village, and has gone as far as telling Mistress Hibbins, the town's alleged witch, that she'd be willing to go in the forest with her to cast a spell on whoever dares to come in between herself and Pearl. Therefore, Pearl is just as willful and determined as both of her parents.
Let us remember, however, that Pearl will change completely once Dimmesdale comes clean and "the curse" is broken. She forgives Dimmesdale, gets married, takes care of Pearl, becomes rich (Chillingworth leaves her his fortune-go figure) and marries well. However, she definitely grows up putting into the open the hidden traits of her parents, who are themselves hiding their secret from the rest of the world.