You are jumping ahead of the project. Way ahead. First things first, is to determine the thermal and mechanical load.
The mechanical power (HP) requirements of a compressor depends on its thermal capacity. You have to figure out by mechanical calculation or find the data from the manufacturer.
Then and only then, you may decide on a) how much electrical power you require, and b) what type of motor and drive you are going to use.
But to answer your question, the simplest and most straightforward method would be a DC motor which is turned completely on/off by a thermostat. More complex control schemes would employ variable speed. But even then, the compressor's thermo-dynamic characteristics must be met. I am not an A/C expert, but know that a pressure differential must be met for the refrigeration cycle to operate. This pressure differential is depending on temperature differentials and motor speeds.