A simple and easy way to remove background noise from a clip is to use a noise gate. This is basically a method used to reject audio above or below a set threshold. It can be very handy if you are trying to remove background noise from a given audio track. Set the lower threshold to something above the noise floor and below the lowest need audio level. Anything below that threshold will be removed from the audio clip, that way whenever someone isn’t speaking into the microphone you have silence.
Adobe hides the noise gate feature under the label “dynamics” while FCP editors can find the feature under the label “AUDynamicsProcessor”. If you don’t use either of these programs and your editing software doesn’t have this feature you can export your audio and perform the same task in free programs like Audacity. For a step by step guide in Audacity you might want to take a look at this.