Toyota Repair Problems? Ask a Mechanic for Answers ASAP
The A/C system operates on both refrigerant pressures and electrical signals - one won't work without the other. As such, you'll need a set of gauges to read refrigerant pressures and a multimeter to check the circuits. Also, in computerized vehicles such as yours, the onboard computer will cut power to the compressor under some circumstances such as high engine temps, misfire codes in memory, etc.
The first thing I would recommend is checking that relay you just replaced and see if it clicks when the A/C is turned on. If it doesn't, then we can take it from there - as long as you have the necessary tools, that is...