China did not affect all three of these countries in exactly the same way. However, it did have some influence on all of them.
In both Korea and Vietnam, a Chinese style of government was adopted. Both tried to have an emperor with a bureaucracy that was just like that of China. Korea had an examination system for picking its bureaucrats just like China did.
In all three countries, Chinese writing was used either exclusively or along with a native script. In Japan, for example, Chinese characters were the first form of writing and a native syllabary was only made later and was based on Chinese characters.