Like any relationship, communication is key when bringing voice talent on to your project. This video is your baby, something you’ve been working on for awhile, so you will get the best results if you can communicate your vision for the project. Provide background on your brand, on your customers, because your voice actor can use these details like fuel to accurately represent your brand.
It's only after this sort of processing that I'll start cleaning up the recording and dealing with the gaps of silence between spoken passages. A noise gate (or an off‑line 'strip‑silence' function) can be used to automatically mute sections that fall below a certain level. However, if not used carefully, these tools will clip the 'T's and 'P's off the ends of words, and shut out natural breathing sounds. Worse, if your recording is noisy, a noise gate will actually draw attention to the problem, since your client will be able to hear the difference between room tone and absolute muting. Another trick is to use a downward expander to reduce the noise floor of quiet sections, rather than cut out the noise completely.
As editor, you need to turn the raw recording into the final version of the file you'll be sending to your client. Editing during a recording session can be problematic, and particularly so if you're recording in a domestic environment. How can you expect to hear the drone of the dishwasher ruining your recording if the dishwasher is still running as you edit? I'll look in more detail at setting up a voiceover recording studio next time, but for now, if recording in a separate room or isolated booth isn't an option, at least try to edit the file the following morning (or maybe at night) when there's less noise around — because hearing your recording with a fresh ear and in a quieter environment will help you to make better editing decisions, and will reveal noises that slipped your attention earlier.