If your fridge is not cooling but you can hear the compressor running, the issue is more than likely with the evaporator fan. The evaporator fan helps move air around inside of your fridge. You can replace it on your own or hire an appliance repair company to replace the fan for you.
#1 Take Off the Evaporator Cover
First, take off the cover for the evaporator unit. This cover is for the entire evaporator system. The cover for the evaporator system is usually located on the back of the freezer. The cover generally extends across the entire back of your freezer if you have a top/bottom style fridge and freezer unit.
There are screws that hold the panel in place you will have to remove with a screwdriver. Oftentimes, the screws are hidden behind plastic plugs you have to pry off in order to get to the screws.
Check your owner's manual in order to find out where the evaporator cover is located on your fridge.
#2 Remove Evaporator Fan Cover
The fan is located at the top of the evaporator unit. Although you may be able to see the fan, there is generally a cover around the fan you will have to remove in order to replace the fan. Use your screwdriver to remove the cover over the evaporator fan area. Removing the cover will allow you full access to the evaporator fan.
#3 Unplug the Refrigerator
Before you remove the evaporator fan, turn off your fridge. You don't want the evaporator fan to turn on while you are replacing it.
#4 Remove the Evaporator Fan
Removing the evaporator fan should be an easy process. Just remove the screws, located on the sides of the fan, in order to remove the fan from the wall of your freezer. Once you remove the screws, pull the evaporator fan away from the wall of the freezer. Unplug the wire power outlets from the evaporator fan.
#5 Install the New Evaporator Fan
Take the new fan and plug the power outlets into the new fan. Attach the fan to the wall of the freezer, then put the fan cover back in place. Finally, put the cover for the entire evaporator unit back in place, and remember to put the plastic plugs over the screws.
Replacing the evaporator fan should get your fridge cooling again. If you need further assistance with fixing your fridge, go to sites of local appliance repair shops for assistance.