Grogan developed her singing career as the lead singer of Altered Images, originally a five-piece band that included Johnny McElhone (later of the Scottish rock band Texas), whom she met while studying for her Highers. It became a four-person band with the departure of two members and the addition of Stephen Lironi, who played both guitar and drums. The band had a string of hits in the early 1980s, including "Happy Birthday", "Don't Talk to Me About Love", "I Could Be Happy" and "See Those Eyes". The group split up after the release of their third album, Bite (1983).

While working as a waitress at the Spaghetti Factory restaurant in Glasgow, she was spotted by film Director Bill Forsyth. This led to her breakthrough acting role in 1981's Gregory's Girl as Susan. Because of her facial wound, there were objections from the producers, but Forsyth refused to recast the role and Grogan was filmed mostly in profile. When filmed in close up, makeup artists covered Grogan's scar with mortician's wax.

In 1984, she played Charlotte in Forsyth's Comfort and Joy. In 1985, she was the receptionist in the BBC Television version of Blott on the Landscape. She had a recurring role playing Dave Lister's would-be love-interest, Kristine Kochanski, in series 1, 2 and 6 of the TV show Red Dwarf. In series 7, she was replaced by Chloë Annett. Grogan has also appeared in Father Ted (episode "Rock-a-Hula Ted") as a feminist rock singer and in EastEnders as Ian Beale's love interest, Ros Thorne (1997–1998).

Grogan later attempted a solo career, but after her single "Love Bomb" failed to gain chart success in 1987, her album Trash Mad was never released. Grogan formed Universal Love School in 1989 with Lironi, performing a series of gigs around the UK. However, it was short-lived and produced no hit singles. In 2000, she contributed vocals to the song "Night Falls Like a Grand Piano" on The 6ths' album Hyacinths and Thistles. She recorded a version of "Angels with Dirty Faces" for the Frankie Miller tribute album. The track "Her Hooped Dream" appears on The Ultimate Celtic Album.

Grogan married band mate Stephen Lironi in Glasgow in 1994. The couple live in Haringey, and in 2005 they adopted a daughter.

She appeared in the 1998 comedy Jilting Joe, played control-freak office manager Sandra Reeves in the 2006 film The Penalty King, and starred in the Scottish gangster movie The Wee Man in 2013. She also appeared in the video for Peter Kay and Matt Lucas' charity single "I'm Gonna Be (500 Miles)", recorded for Comic Relief's Red Nose Day 2007. She also played Danny's mother in the 2006 Scottish sit-com, Legit.

In 2002, Grogan performed as 'Altered Images' on the Here and Now Tour which featured other well known artists from the 1980s. She performed on similar tours in 2005, 2008 and 2009. She appeared with Chesney Hawkes, Toyah Willcox and Limahl as 'The 80s Supergroup' in the 2011 series of Let's Dance for Comic Relief.

Grogan's first book was published in October 2008, a children's novel (aimed at age 7 and up) titled Tallulah and the Teenstars, about a girl who forms a pop band.

Clare appeared in Series 2, Episode 2 of John Shuttleworth's Lounge Music on BBC Radio 4 on 27 November 2016.

Grogan began presenting shows on Absolute Radio 80s from 11 September 2017, presenting Monday to Thursday 8-9pm and Sundays 7-9pm.