Short answer: normally transliterated Russian names cannot start with C
.
Long answer: sometimes the name bearer can change that.
There is a law (Приложение N 6 к Приказу ФМС от 26.03.2014 N 211) that explicitly states how Cyrillic names should be transliterated in Russian-issued foreign passports. However, it also states that the applicant may have their name registered in another transliteration variant if they provide an official document (e.g. a passport issued under older rules, by the foreign country's authority etc).
Also, some people prefer to use more "westernized" variants of their names (especially in social networks):
Константин → Constantin (instead of Konstantin)
Клавдия → Claudia (instead of Klavdiya)
Клара → Clara (instead of Klara)
This may also apply to foreign citizens of Russian descent. One notable example is Russian-Estonian Климент Иванов (Clement Ivanov).
Initial C
aside, there are also other interesting cases. Famous boxer Владимир Кличко spent most of his career in Germany and was promoted there, so he chose that his name be written in German translit: Wladimir Wladimirowitsch Klitschko. This variant passed on into all official sources. A counterexample is another famous boxer Костя Цзю, known internationally as Kostya Tszyu, which is closer to traditional Latin transliteration of Russian. As he is of Korean descent, he may have chosen to have it spelled Ju (주), but obviously preferred not to.