In this first clip you can hear the audio in its camera original form. Lot's of audio hum and crackle - yuck.
So the first task is to try to eliminate the hum. Premiere Pro includes a notch filter to correct such problems. (A notch filter passes most frequencies unaltered, but attenuates those in a specific range to very low levels.) The key to successfully applying a notch filter is to identify the frequency of the offensive hum. The easiest way to do this is to apply the filter and then slowly sweep through the frequency range, previewing the effect after each adjustment. In this case the hum was predominantly located at 150 Hz.
That improved things a bit, but notice there is the hum is also present at a higher frequency range. The solution is to apply a second notch filter using the same technique. In this case the hum was present at about 310 Hz. Listen to the next example and think you'll notice that the hum is gone.
The next problem you may notice is the persistent crackle in the audio. It seems that the wireless mic system used to record the audio added this crackle as well. Premiere Pro includes an easy-to-use filter for this problem aptly named "DeCrackler". Like the notch filter, you can preview the filter will have on your audio and tweak the settings accordingly. In this case, the Threshold was set to 2% and the Reduction was set to 94%
Finally, there's a lot of airy noise present. Sure enough, there's a filter to correct this too - DeNoiser. In this case, I reduced the hiss 5dB. In this next clip you'll hear the cumulative effect each of the filters described above.
Here's a still which shows the filters that were used to fix this audio.
One thing to keep in mind when applying audio filters is listen to ensure the filter degrades your audio as little as possible. The point is to attenuate or remove the offensive noises without harming the sound you want to preserve.
To learn more about working with audio, check out Jay Rose's Audio Postproduction for Digital Video.