出生於台北市大稻埕，在被列為市定古蹟的榮星幼稚園接受音樂啟蒙。奧地利維也納市立音樂院演奏家文憑特優獎（Auszeichnung）畢業，美國琵琶地音樂院大提琴演奏碩士，曾隨已故大提琴泰斗羅斯卓波維奇（Rostropovich）前往法國埃維揚（Evian）音樂節為其委託製作之作品做世界首演，為臺灣中生代最受矚 目的大提琴演奏家之一。

留歐美十數年，前茱莉亞絃樂四重奏小提琴家Earl Carlyss稱她的演奏「音色流 暢優美…結合了高度智慧、深度感性與原創性」；皮亞提果斯基（G. Piatigorsky）嫡傳弟子大提琴家Stephen Kates則說她「天資聰穎、敏銳而積極」。返台後獲選為年度樂壇新秀，並陸續於北藝大、輔大、交大、高師大等音樂系所展開其教學生涯，並活躍於獨奏、室內樂與當代音樂的各項演出。曾應邀於亞力山大·魯丁（Alexander Rudin）、胡瀞云等國際知名音樂家來台之音樂會中合作；為英國作曲家Dorothy Kerr作品《Solo for Cello》之詮釋，深受作曲家本人喜愛而指定於BBC英國國家廣播節目中播出。

劉姝嫥近年曾參與奇美博物館「提琴數位典藏計畫」及國立傳藝中心「臺灣音樂憶像」之錄音，並持續致力於國人作品的演出。除曾為吳丁連、李子聲、王思雅等臺灣作曲家之大提琴作品世界首演外，率學生成立之大提琴重奏團（C.C. Liu's Cello Ensemble）也曾委託作曲家顏名秀創作為八把大提琴的《客家隨想》。 2017年起，於義大利克里蒙納國際音樂節中（Cremona International Music Academy）任教並演出；2019年，於法國里昂舉行室內樂大師班並擔任德國杜超威大提琴大賽（Dotzauer Competition）評審團亞洲區代表；2020年，與長榮交響樂團（ESO）於關渡藝術節協奏，並於貝多芬誕辰250週年與蔡佳憓教授演出全本貝多芬大提琴奏鳴曲；2021年再度與長榮交響樂團合作於國家音樂廳演出舒曼大提琴協奏曲，2022年與詠嘆調大提琴四重奏巡演德佛札克的大提琴協奏曲，並於衛武營音樂廳演出黎俊平的大提琴協奏曲《 落葉歸根》。

Chu Chuan Liu is one of the last pupils of the great Austrian cellist Senta Benesh at the Hochschule fur Musik und Darstellende Kunst Wien (Vienna), and received her artist diploma from Konservatorium d. Stadt Wien with highest honor(Auszeichnung). During the time in Europe she has also performed in master classes for several renowned cellists: Michel Strauss (Paris), Heidi Litchauer (Salzburg), Christopher Henkel (Freiburg) among others, and for a short period of time, studied Viola da Gamba with Jose Vasquez and participated in chamber music seminars lead by the Wiener Schubert Trio.

In 1993 LIU went on to study with Stephen Kates at the Peabody Institute on a scholarship. As member of the Peabody Camerata , she then returned to Europe to perform under the baton of Maestro Rostropovich at the festival of Evian, France in 1994.

She was awarded "Young Musician of the Year" by the cultural minister as she went back to Taiwan in 1995, and started giving concerts both as solo and chamber cellist. Once a very enthusiastic performer for contemporary music, she has appeared with CCOT (Contemporary Chamber Orchestra Taipei) in music festivals in New York, Boston, Vancouver, Shanghai, and gave world premieres for solo works by Asian and European composers alike.

Chu-Chuan Liu is cellist of the Po-Ya Piano trio and has been invited to collaborate with chamber music society of Minnesota and cellist Alexander Rudin, violinist Chao-Liang Lin, pianist Ching-Yung Hu at their chamber concerts in Taiwan. Currently associate professor at the Taipei National University of the Arts – a forefront institution of its kind in Asia, LIU is also founder and artistic director of CCliu's Cello Ensemble.