I registered my details on the main Skype page via my PC.
I also downloaded the (free) Skype software to my PC, this way I can make the profile amendments etc.
Open the Kindle Fire HD and begin the setup - A click should do it (no need for you to download any software the Kindle does it itself).
As you've registered already input your user details.
Skype will then launch with your setup and friends.
Click Contacts and select a friend. It will give you a list of options (inc Video Chat).
What I did as it is for my daughters Christmas present was to set myself an account on the PC for me, and initiate a video chat to see if it worked. It did
Hope that helps.