She was named, at her father, Mitch's insistence, after her great aunt, Mitch's aunt Julie. Mitch admired his aunt for her bravery and her compassion, and also owed her for practically raising him from childhood. Natalie would later tell her daughter that the name suited her very well.
She briefly went to the Sea of Tranquility exhibit at the science museum, and bought an aquarium from the gift shop for Mr. Henry, not realizing that the aquarium in question actually contained a priceless moon rock that was stolen by one of the tour guides at the museum.
In a manner similar to Benjy Fleming, the son of Monk's prior assistant, Sharona Fleming, Julie was initially unsure what to make of Monk and his several quirky habits. But she was soon impressed by his deductive abilities (to quote her, "he's just like Velma from Scooby-Doo"), and then bonded with him after he rescued her pet fish, Mr. Henry (a gift from Mitch), from death - an act which also won Natalie's loyalty as his new assistant. She also never knew that the original Mr. Henry actually died six years prior.
Although she is easily exasperated by some of Monk's more eccentric habits, she comes to accept him as a surrogate member of their family, including agreeing to go trick-or-treating in Tewkesbury while Monk was visiting his brother, Ambrose, and allowing Monk to spend the Christmas holiday with them.
She inadvertently became caught up with one of Monk's investigations when she was noticed by fashion designer Julian Hodge and asked, despite her young age, to model in one of his shows. She was ecstatic, to the point where, when Natalie forbid her to do it because Monk suspected Hodge of murder, she ran away from home to be in the show. But when Natalie confronted her on stage, she immediately felt sorry and agreed to go home with her mother.
Having seen his detective work in action enough times, Julie and her friends from the high school basketball team hired Monk to investigate when their coach was found dead in the locker room. They (rightfully) suspected that this was no accident. In addition to solving the murder, Monk briefly served as assistant coach to Natalie, filling in for the deceased coach.
An important bonding moment for Monk and Julie came when she was confused about her choice to break up with her current boyfriend, Tim, in favor of a dashing football star who was interested in her. For all his eccentricities, Monk knew more than a little about love, through his relationship with his wife, Trudy. Julie responded to his advice, and recognized which boy was right for her - a decision reinforced when Monk exposed the football star as a fraud put up by a killer, who was trying to mislead her and Natalie. Tim played a heroic role in saving the evidence that helped convict the killer, and won Julie's heart back.
Natalie admits that she has trouble saying no to her daughter; Monk also learned this when Julie begged to take her scheduled driver's exam, despite the fact that a killer seemed to be stalking women who shared her name.
She is seventeen years old (as of Season 7, Episode 3: Mr. Monk Gets Lotto Fever) — as per Natalie as she reads the lotto number '17'.
In the Season 8 episode Mr. Monk and the Critic, Julie performs in a theatrical production and sings "Away". At the end of the episode, she learns she has been accepted to the University of California at Berkley.
In Mr. Monk and the End, Part 1, Julie mentions to her mother, Adrian and Steven Albright that she is planning to move to Berkley when she starts college. Julie appears in flashbacks in Mr. Monk and the End, Part 2.
Full Article: Mr. Monk (book series)
Julie appears in several of the several different books in the Mr. Monk novel series though in the first few novels, Julie's role is minimal, but she occasionally, knowingly or not, gives Monk crucial information that is key to solving a murder case.
Behind the Scenes
Julie was likely named after Monk creator Andy Breckman's daughter Julie.