If you hear the compressor running but the freezer doesn't cool, a sealed system problem such as a refrigerant leak or a failed compressor pump may be causing the problem. A service technician will need to fix a sealed system failure because the repair involves the recovery and recharge of the refrigerant.
If the freezer cools but doesn't cool down to 0 degrees, a failed thermistor may be inaccurately detecting freezer temperature or the electronic control board in the freezer may be inaccurately detecting the thermistor signal. Using the freezer's tech sheet as a guide, unplug the freezer and check the resistance of the thermistor using a multimeter. Replace the thermistor if its resistance differs from its expected resistance. If thermistor resistance is okay, check the resistance of the thermistor circuit from the wires that connect the thermistor circuit to the electronic control board. If you measure the same resistance through the wires that connect to the control board as you measured directly at the thermistor, then you'll need to replace the electronic control board because it isn't accurately detecting the thermistor signal. If the thermistor circuit resistance differs from the thermistor resistance, then a wiring failure in the thermistor circuit is interfering with the thermistor signal going to the control board. Find and repair the bad wiring. If your freezer uses a dial control, the cold control thermostat may not be sensing and controlling freezer temperature properly.
Problems in the automatic defrost system can also prevent the freezer from cooling. You'll typically see frost on the evaporator cover when the defrost system isn't working because the defrost heater isn't turning on to melt frost off the evaporator fins. The evaporator won't be able to cool the freezer well when the evaporator cooling fins are caked with frost because the frost blocks air passages over the fins. To check the defrost heater, unplug the freezer and check for continuity through the defrost heater element using a multimeter. If you don't measure continuity through the defrost heating element, replace it. If you measure continuity through the defrost heater, it's likely okay.
Check the defrost bi-metal thermostat next. A failed defrost bi-metal thermostat won't allow current to reach the heater. You can check for continuity through the defrost bi-metal. (Note: When the bi-metal warms up, it trips open to shut off the defrost heater to prevent the heater from overheating and damaging the evaporator so you can't check the bi-metal when it's above 15 degrees.) If you don't measure continuity through the bi-metal, replace it because it isn't allowing the defrost heater to turn on.
A failed defrost timer won't advance into the automatic defrost mode so the defrost heating element would never turn on to melt frost off the evaporator fins. You can check to see whether the defrost timer advances by manually advancing the defrost timer to the defrost cycle. Turn the dial on the defrost timer with a screwdriver until the compressor shuts off. If the compressor isn't running when you're ready to test the defrost timer, open the freezer door until the compressor turns on and then advance the defrost timer until the compressor shuts off. The defrost cycle takes about 30 to 45 minutes to complete then the compressor should turn on to cool the freezer. If the defrost timer doesn't advance through the automatic defrost cycle and allow the compressor to start after 30-45 minutes, it's likely the timer is broken and needs to be replaced.
A failed evaporator fan can also cause cooling problems because it won't circulate air through the freezer cabinet. Replace the evaporator fan if it doesn't run when activated.
Clean dirty condenser coils because dirty coils won't cool down refrigerant before it enters the evaporator, resulting in poor cooling performance. Replace the condenser fan if it isn't running when the compressor runs because the condenser fan blows air across the condenser to help cool the refrigerant before it enters the evaporator.