Hyundai Repair: A/C, mechanical failure, expansion valve

Question
I have a 04 Elantra and about a month ago I went to turn my a/c on and it worked fine for about 5 miles then it started blowing hot air.  The fan was working, so I thought it just needed refrigerant.  My dad got a self fill kit for me and when he went to do it the gauge said I had enough refrigerant but I might have a mechanical failure.  The a/c will blow cool, not cold, for about a minute before heating up.  I am wondering if it is more likely to be the compressor or the condenser.

Answer
There are a number of components which could cause the A/C to not work if the refrigerant level is proper.  The condenser is not one of these components, since its primary mode of failure would be leakage.

Since the system blew cold immediately prior to blowing cold, we can probably rule out the blend door in the heater box that regulates whether you're receiving heat from the HVAC system.

I see the following as reasonably likely potential problems:
-- Faulty expansion valve
-- Faulty evaporator thermistor
-- Faulty pressure switch
-- Faulty compressor
-- Faulty A/C relay
-- Engine overheating