Whatever the origins of the Communist Party in Gotham may have been, the modern party is a wholly owned subsidiary of Josef Stalin's Soviet State. Whatever is good for Stalin is whatever the Communists advocate, and if that changes tomorrow, the party line changes tomorrow.
The goals of the party currently are to gain control of the Dockworker's Union, a move which is bitterly opposed by various criminal gangs who already make a nice living off it, and to press forward the unionization of the Fashion District, simultaneously gaining control of the Garment Workers Union, a move particularly resisted by the Falcone Family. The fashion district operation was dealt a severe setback recently when the cell in charge was wiped out by masked vigilantes.
The party maintains a public Command Post in Gotham Heights
Some more recent Chinese immigrants are sympathetic to Mao's attempted revolution and have begun to warm to the local Communists.