There is no way you can use a pump to produce a refrigeration cycle. You will need a compressor.

To do any refrigeration you will need to vaporize the refrigerant and then compress it and condense it into a liquid and repeat the process. To select a compressor you need to know at what temperature you are going to try to condense the refrigerant because that will determine the discharge pressure. With the suction pressure and discharge and the other refrigerant variables such as the specific heats you can determine the horsepower for the compressor driver.

As previously stated you will also need a condenser and an evaporator (chilling coil).

You might want to look into an absorption refrigerantion system. See below.

Do you want the refrigerant to evaporate to carry the heat to the condenser? Or are you just pumping liquid around to take heat from a hot place to a cool place. You didn't mention an evaporator or condenser.

One way to get a compressor would be to cut one out of an old refrigerator that is sitting at the curb on trash pickup day.