When China people's names are written in Japanese, should they be written in Katakana?

No, you don't have to write it in katakana. It doesn't matter if the words in your name are not in Japanese.

Most Japanese surnames and names are expressed in Japanese characters, and some surnames are Chinese characters with pseudonyms, such as Ayumi Hamasaki. Or the surname is a pen name and the first name is a Chinese character; Either under a pseudonym or in English. Japan's name management system is relatively loose.