Skype is showing up in Launchpad twice — why? How do I fix it?
Well, a bit more research and I may have found a solution. 

* Proceed with caution. Backup files. Be smart. This worked for me YMMV. *

The Easy Solution

The most straight forward way to fix this problem seems to be to delete the SQLite database that stores Launchpad's data. To do this simply open the following folder,

~/Library/Application\ Support/Dock


and delete any .db files you find in there. They'll have a file name that looks like this,

58357ED0-F8D5-4EB3-9A1F-E6B71B0A001D.db


Then open terminal and type this command followed by return:

killall Dock


But be careful, you will lose any icon groups you created in Launchpad. I didn't like this solution for that reason so I while this worked to get rid of the duplicate icons I used the following solution on my machine.

The Hard Solution

I opened the above mentioned folder and located the above mentioned file and then opened that file in SQLite Database Browser by dragging the file onto the icon for the db browser.

Once I had the file open I clicked Browse Data and then selected Apps from the Table menu. 



I then scrolled through the list of apps until I found one of the duplicate records for Skype and selected it.



With the duplicate record selected I clicked Delete Record.



I then quit SQLite Database Browser and acknowledged that I wanted to save my changes. 

Then I switched to terminal and typed this command followed by return:

killall Dock


And Skype's doppelganger was thereby vanquished from my Launchpad.