Why would my ac not blow cold air? Took it to a mechanic which said it was the alternator (which was going out) so paid to put in a new one,but there problem is still accruing. He said he may have to go thru the dash to find the problem.

Standard A/C diagnostic in our shop would be to evacuate the refrigerant to verify there is enough in the system. If the system has the correct amount of refrigerant, then we re-charge and check pressures and component functionality. If they system is low on refrigerant, you have to figure out why, as it has a leak in the system somewhere. This is usually accomplished by putting a dye back into the system with the refrigerant to detect where the leak is coming from. All that said to show that there isn't an easy answer to your question. There are several components to a working a/c system and would most likely need to be diagnosed by an a/c technician to find the definitive answer.

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Is the compressor turning on at all? First question , then IF yes , without spending money on evac & recharge , a simple check of the 'static' pressure and knowing the correct ambient temperature can give you a pretty good idea of state of charge. Then IF NEEDED recharge system. You did not specify any details as to if it was cooling a little or not at all. I recently had a customer that had been told by 2 different shops , that his air-mix door was most likely the problem- his symptom was it would cool on the passenger side properly , but on the drivers side it would mostly start off slightly cool , and then warm. He requested diagnostics relating to that when he brought it to where I work. I took 5 minutes (literally) to bring the car in and hook it to our A/C machine and check static pressure. It was 55 PSI in an environment where it should have been closer to 90 , I recharged it with dye to monitor for leaks , and it worked beautifully! AND I didn't need to tear his dash apart needlessly ---moral of the story , find a shop you trust , with COMPETENT service personnel.