What I mean is people forming an organization that provides health care, education, food to people, and acts like it is the government? I think of it as a (good) parasite that slowly pushes a greater government out of it's relevancy to society, to the point where it dissolves. Are there any works from Marx or Lenin that mention something like this? I'm starting to read about Leninism.
What you're describing is "dual power," but that doesn't begin until, in the first place, definite economic power has been wrest from a significant section of the bourgeoisie via soviets/councils, and second the existence of the vanguard party to lead these soviets/councils towards the overthrow of the capitalist state. That was to an extent the situation between the February and October Revolutions, in which the Provisional Government and the soviets were at odds with each other.
How would those Soviets come into power?
By armed revolution, obviously.
Not that I'm well versed in its history, but did the Black Panthers not try attempt something similar with its Free Breakfast for Children program?
Yes, I don't know the details of such a program but I know they did push for and give children a free breakfast before they went to school. http://marxists.org/history/usa/work...1969/03/26.htm