What do you use for a microphone? I have a Shure SM57, bought to record guitar, that I have repurposed into my PC mic, and the reduction of background noise vs laptop/webcam mic is _enormous_. You can't hear my mechanical keyboard at all, and it's sitting about a foot from the mic. The SM57 isn't really a vocal mic, but it works great. I'm sure a more purpose built vocal mic would be even better.

If you are currently using a built-in mic then I think something like that in conjunction with a BG noise reducer would basically get you the rest of the way. You just need to figure out a configuration that works for you and keeps it (at least relatively) out of sight.

It also has the added benefit of hugely improving your audio quality.