How can an American become a Chinese citizen?

A foreigner has to apply under article 7 of the National Law of the People's Republic of China. To qualify under the article, a foreigner must have close relatives who are Chinese nationals, have settled in part of China, or have other legitimate reasons for applying for citizenship.

Go to Hong Kong and wait for the military to force you into the custody of the mainland. 

Convert to communism and pledge your allegiance to the Chinese Communist Party and their founding fathers.