Clare is the daughter of California University's Chancellor Milton Arnold. She was largely introduced as a privileged, bold temptress who was used to getting what she wanted. Pre-show, she grew up in a fictional European country where her father was an economic consultant. Her mother was best friends with the mother of Prince Carl, her childhood friend who Clare described as "a spoiled rich pain the butt".
Clare's mother passed away at some point while Clare was still young and she and her father ended up in California. She was introduced at a retreat for the Chancellor's task force, where she roomed with Kelly Taylor. Who Clare initially wanted was Brandon Walsh, and she wasted little time in making this clear. During an evening in which the two of them had wound up alone at his home, she made a series of playful passes at him. Brandon, being a CU student at the time, felt uncomfortable due to the position that Clare's father held, and soundly resisted. Claire later threatened to get Brandon in trouble with her dad, but none of her threats were carried out, and by the time Season 5 rolled around Claire told Brandon that she was sorry about having acted like such a psycho, leading to her being accepted by the rest of the group.
She later moved in with and befriended Kelly Taylor and Donna Martin at a beach house apartment. It was during this period that she also began seeing David Silver. An early point of their relationship involved her catching him handcuffed to a bed, and deciding to have her way with him. They were interrupted, however, when her father suddenly arrived outside, causing Clare to leave a much-agitated David alone until he was released later on. They also made a X-rated video of each other that was nearly viewed by Donna Martin's parents (though Dr. Martin strongly implies he saw the whole thing and found it amusing). While there was some initial turbulence between them over Claire not telling her dad they were dating, things went downhill in Season 5 when David admitted he still had feelings for Donna, leading to an initial breakup, a brief reunion, and then a final parting of ways between Seasons 5 and 6.
After she and David had broken up, Clare began dating Steve Sanders after she became his math tutor. Though both were playful, it was during this relationship that Clare began to reveal a more serious and sensitive side as she and Steve fell in love. However, at the end of their first year together, Prince Carl came to California determined to win Clare's love. He was ultimately unsuccessful and Steve and Clare remained together despite his interference.
On one occasion, Clare was able to overcome stage fright by looking out at Steve and directing a song towards him. In another episode, the couple briefly addressed the topic of marriage and where they each stood on it, eventually deciding that they'd simply follow their hearts. As Season 7 wound down, Steve's uncaring mother dumped Claire's father, and Claire (who had spent much the year pissed off at Steve for various things he either did wrong or she inaccurately blamed him for) pushed Steve away once again.
Following their college graduation at the end of the seventh season, Clare reluctantly left Steve—and subsequently, the series—when she accompanied her distraught father out of town, wanting to ensure that he didn't feel alone. Her last on-screen moment was with Steve on the beach.
Clare had an obsession with Brandon that was never reciprocated. Going so far as to handcuff herself to his bed and also tell her father that Brandon was taking her to prom, Brandon did everything to discourage the infatuation. The two eventually form a friendship but Clare always harbored a crush on Brandon.
David and Clare start dating in season 5 and even though they had a relatively smooth (if uneventful and occasionally immature) relationship, it wasn't long before they break up and move on to others.
Steve was Clare's longest relationship but her obsessiveness over everything from her schoolwork to her father to her deceased mother drove a rift between them. While they stayed together until she left Beverly Hills, the relationship was under constant strain from Clare's moodswings over her protectiveness of her father and the constant depression over her mother no longer being in her life. While Steve made an effort to please Clare most of the time, she ultimately chose to follow her father to France and break up with Steve for good.