Before deleting your Skype account, you must first delete the personal data of your Skype profile.
- Click on the Skype menu > Accounts
- Scroll to "Account details" and click on "Edit Profile":
- Go to the Personal information and click on "Edit".
- From there delete or replace your personal data (replace name by xxxxx).
- Click on Save.
You can permanently remove your name from Skype, by making a request here.