The ATP rated pilot was starting a scheduled Part 135, cargo run, departing on runway 19, a 5,000 x 50 foot, asphalt runway. The runway was covered by one inch of snow and a snow shower was in progress. The pilot said he did not remove snow from the aircraft prior to the takeoff attempt, because it did not appear to need it. The airport is neither approved nor equipped for instrument operations as required in the FAA approved company operation specifications. The pilot reported the visibility as '2 to 3 miles,' with calm winds and said he could not estimate a ceiling through the falling snow. Three FAA inspectors, who were 3 miles from the airport, said the visibility was less than 2 miles, and the ceiling was indefinite, less than 1000 feet. An IFR flight plan was filed. During liftoff, the right main gear entered deep snow on the right edge of the runway. The aircraft settled back to the ground and pivoted 270 degrees to the right. The nose and left main landing gear collapsed as the aircraft stopped.