I am not aware of the correct pronunciation, but as far as I know, in the Southern Germany and in Austria they say 'Kina' (/k/).
The other German say it as it is: 'China' (ch as /ç/)
Some region (for example where I live), some people say 'Schina' (/ʃ/), but they also say 'isch' instead of 'ich' or 'Mädschen' instead of 'Mädchen'. That is, we pronounce the 'ch' in a different way, but that's not necessarily wrong, though no standard German.
The best way (imho) is to say /ç/ as in 'ich', then everyone will accept it.
Edit:
I found a wikipedia article giving some pronunciation rules. Amongst other things it says:
Außerdem kommen für ch auch die Aussprachen als sch [ʃ] und tsch [tʃ]
vor:
wie [ʃ] als umgangssprachliche oder varietätenspezifische
Variante von [ç] am Wortanfang (Chemie, China),
in nicht (vollständig) integrierten Fremdwortschreibungen vor allem
aus dem Französischen und Portugiesischen (als [ʃ]: Champignon,
Charlotte, Recherche), dem Englischen und Spanischen (als [tʃ]: Chip,
chatten, Macho).
This means it is acceptable to pronounce it as /ʃ/.
As summary:
You can say China, Schina and Kina, but in case of this country there seems to be no 'Tschina'-pronunciation, although it is possible for some other words starting with 'Ch'.
An additional information: The same is valid for 'Chemie', but NOT for 'Chemnitz' and 'Chiemsee'. Both are spoken with the K-sound.
I want to link an article from belleslettres.eu about the right pronunciation of the word Chemie. The most important thing he imho mentions is:
Welche Aussprache Ihnen auch immer eigen ist, Sie liegen damit richtig. Denn für die drei Varianten /ʃ/, /ç/ und /k/ finden sich jeweils so viele Sprecher, daß man von keiner Standardform durch Mehrheit sprechen darf.