When do you contact ATC when departing from a non-towered airport into class C controlled airspace?
This is spelled out clearly in CFR 91.130:


  (c) Communications. Each person operating an aircraft in Class C airspace must meet the following two-way radio communications requirements:
  
  (1) Arrival or through flight. Each person must establish two-way radio communications with the ATC facility (including foreign ATC in the case of foreign airspace designated in the United States) providing air traffic services prior to entering that airspace and thereafter maintain those communications while within that airspace.
  
  (2) Departing flight. Each person—
  
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (i) From the primary airport or satellite airport with an operating control tower must establish and maintain two-way radio communications with the control tower, and thereafter as instructed by ATC while operating in the Class C airspace area; or
  
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (ii) From a satellite airport without an operating control tower, must establish and maintain two-way radio communications with the ATC  facility having jurisdiction over the Class C airspace area as soon as practicable after departing.


You can try contacting them while on the ground, but if you can't raise them, you have to do it as soon as you can after take-off.