It seems unlikely that using Skype from another computer would have any effect at all on your Kindle Fire HD. I use Skype from my PC (a Dell XPS 1645), my Kindle Fire HD 7", and my cell phone (Droid RAZR). It works well on both my computer and on the Kindle Fire, and I've never had a problem getting the video to work. I have other issues with the way it works on my phone, and have never tried a video call on my phone.
I noticed a comment in the Amazon app store saying that "[Skype] works well now that it's installed correctly" or something to that effect. Maybe uninstalling it and reinstalling it would help?
The only other issue I can think of is maybe you have an "interesting" network connection, or maybe you're expecting video when there is a reason why there might not be any (voice call, or call to a normal phone)? Or maybe you're Kindle/Skype is working fine and the other end of the call is having issues?
Have you tried using a camera app to see if the Kindle camera works at all?
And, if this is a Kindle issue, the question is: what can we do for you to help get you better support from Amazon?
I am going to try removing my contacts then reinviting them to see if that fixes the issue. I wish skype had a way to try the video like they do the audio. This would help in troubleshooting situations.