My deep freeze keep running. Does not shut off or freeze. What could the problem be? What are some tests I can run?

No cooling at all points towards a sealed system problem.

To check this ensure that the compressor is running all the time except when the unit goes thru a defrost cycle.
You can usually hear it but also should be able to feel it vibrating.

If you get some cooling then it may be a defrost problem.
Remove the cover in the freezer and check the evaporator/freezer coils.
If they are clogged solid with frost/ice then it is a defrost problem.
This may be the defrost timer, defrost thermostat or defrost heater.

If you get some cooling and there is just frost in one area (usually where the freon enters the coils) then is is probably a sealed system problem. Be sure that the unit has run for at least an hour. If you do this check shortly after or during a defrost cycle you could come to the wrong diagnosis.

If it is a sealed system problem it will be an expensive repair as you need a pro.
Often it makes more sense to put the dollars towards a new unit.