Snow and frost build-up are on the evaporator (or evaporator cover).

Visual inspection of the evaporator itself provides definitive knowledge of the overall condition of a refrigerator. An evaporator in a properly operating system will be at 15°F below zero and be sticky to the touch. After running a few hours, it will be covered uniformly with a thin fine frost. In contrast, an evaporator in trouble will be:

Marginal Temperatures
Some Cooling
Troubleshooting Refrigerators
Diagnostic System