This could be one of several different things actually. Generally speaking when a refrigeration system begins to ice up at the evap coil I tend to look at the defrost control, temperature control, drain line or refrigerant charge. Since you mentioned that the unit had iced up I would start with the defrost control. If it has failed internally it would not allow the unit to cycle the compressor. To test this you would need, at the very least, an electrical meter and some asic knowledge of how to use it. Same with the testing the temperature controller. As far as most residential/light commercial appliances go, the condensate drain will empty underneath the unit and use the heat removed from the refrigerant to evaporate it. Checking the refrigerant charge is a lot more involved.
Most residential units do not supply access ports to attach a set of refrigerant gauges. To this this requires specific knowledge and tools. Not to mention that working with any refrigerant is illegal unless you are certified by the EPA.
Another possibility is that your compressor has failed due to the excessive ice build up. This can and will cause the liquid refrigerant to return to the compressor and "slug" it causing internal damage. At the better end of this scenario, your capacitor could have failed which is a much easier fix.

I hate to nit-pick but that is actually backwards, the refrigerant flows from the compressor TO the condensing coil.