We all think our voices sound amazing and rich... until you hear a recording of yourself. If you're awkward around your digital vocals, there's really only one way to get comfortable -- through practise.
There are myriad ways our voices are recorded these days, from a simple Skype call or Google Hangout, to intentionally putting yourself in front of a crowd via YouTube or Twitch. Heck, you might clam up making requests of Siri or Cortana.
While confidence is key and takes time to build, there are a couple of steps you can take on your aural (and oral) journey.
#3 Fake It Till You Make It
In the short term, there isn't much you can do other than getting on with it. Talking over a microphone to someone is certainly less personal, but it's still a conversation, especially with video chat.
Stop thinking of it as a performance or presentation, which should help relive pressure you might be putting on yourself. Need a distraction? Concentrate on enunciation and volume.
Not sure what you're going to say or how to phrase it? Then stop for a few seconds and consider your next sentence in full. This gives your brain a chance to "edit" your speech, just like you would an essay or letter.
If you're recording and have the benefit of post-editing, that's even better -- you'll have the chance to pick out your finest work and if you're up for it, getting critiques from friends.
#2 It's All About Quality Tools
You already have a false impression of your "analogue" voice, so don't make the same mistake with your digital one. In terms of hardware, there are good options no matter your budget, from studio-level, to the less ambitious.
Also be mindful of how you record. If disk space isn't a concern, then just use a raw format, such as PCM (WAV files for the average user).
Otherwise, there's nothing wrong with using MP3 or Vorbis at high bitrates (320Kb/s is very safe and some would argue overkill). If you'd rather avoid lossy compression, then FLAC is the best choice.
If you're concerned about VoIP apps, including the likes of Mumble / Murmur, don't be. Many make use of the excellent Opus codec, which does a great job of preserving audio quality in bandwidth-limited scenarios.
Bad compression may have been an issue in the past, making even the most baritone of speakers sound like a chipmunk, but that's not the situation today.
High quality audio also has a strange way of making poor video "feel" better, so it's worth getting this right if you have the option.
#1 Listen And Learn
Now that you have easy access to the least-doctored version of your voice, it's time to get listening. Figure out why you're uncomfortable with how you sound, beyond stage fright or anxiousness.
From here, you can refine your voice. It could be as straightforward as slowing down, or avoiding the use of crutch words like "um" and "so". If it's the actual sound you have an issue with, you may need to identify the natural "mask" of your voice.
Still, even the act of slowing down can have the side effect of making your voice sound deeper, so start with the basics before making a conscious effort to "bass" up.