Connect the two 1/4-inch ends of your cable to the outputs of your mixer. The outputs will be labeled "Line Out," "Line" or something similar. Some mixers may have RCA line outputs labeled as "Record Out," or "Rec Out." If this is the case, you may wish to use these outputs, but you will need a dual RCA to mini-plug cable instead of a dual 1/4-inch to mini-plug cable.
Connect the 1/8-inch mini-plug end of your cable to your computer's sound card's input. This may be labeled as "Line In," "Mic In," or simply "Line."
Use an additional dual 1/4-inch to mini-plug cable to connect the audio output from the sound card to an input on the mixer if you wish to be able to control audio-monitoring levels from your mixer board.
Connect microphones to the mixer as normal, and use the mixers controls to adjust input and output levels. Record the audio using your preferred recording software set to use the inputs and outputs of your sound card.