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Is the compressor and the condenser/compressor fan running?
You can usually hear the compressor running. You should also be able to feel it vibrating and it should be warm.

Is the evaporator (freezer) fan running?

If the compressor and both fans are running, remove the evaporator cover from inside the freezer so you can check the coils.
If they are heavily iced up you probably have a defrost problem.
If yes do not defrost before finding out what is causing the problem.

If there is no frost or only frost in one area you probably have a sealed system problem.

Sounds like you may have a sealed system problem.
Here is a good link.
http://www.applianceaid.com/refrigerator.html

Check out #20 and the example pictures part way down the page.

If it is a sealed system problem it will be a bad compressor, a blockage in the sealed system or low freon. All of them will be an expensive repair as you need a pro and there is no guarantee that the unit will be fixed.
It is usually best to put the dollars into a new unit.

If it is a defrost problem the evaporator coils have to be iced/frosted over to the point where the evaporator fan cannot pull air through the coils.

I would confirm whether is is a defrost or sealed system problem by doing a manual defrost.
Then put the evaporator cover etc. back in place and let it run for a couple hours. If you then only have frost in that one area it is a sealed system problem.

If there is frost evenly covering all the coils just get back to us and we will go through how to check the defrost system.
OR
Just search defrost problem in this forum.

Unfortunately it sounds like a sealed system problem.

If the unit keeps running continuously but cannot get down to the correct temperatures then I would replace the unit.
Note: It may shut down for half an hour or so but this will just be it going through a defrost cycle (all fans and compressor will be off)

If it was a defrost problem the unit would get down to the set point temperatures and then start cycling and running correctly until the ice/frost build up became heavy enough to cause problems usually a week or so.