Approaches I suggest you to try in your home computer:
Before starting skype:
Create another user in your computer, for test purposes. It will have a clean environment perfect for testing. This will rule out any broken settings in your account. Log in as this user, start skype, and see if it works.
Make sure your webcam is working fine, using annother application. I suggest cheese.. That will rule out any hardware issues. Also test the webcam using the test user you created.
Delete the ~/.Skype hidden folder in your user's home folder. It will reset your preferences, but it may also fix the bug.
Now start skype. It should probably work.
If not, please post any relevant logs and/or messages and I'll help you out.
Good luck :)