Summarize the following:
Go to https://www.skype.com/ in your browser. If you're already signed into Skype, you'll see your username in the top-right corner of the page. If you aren't already signed in, click Sign in in the top-right corner and enter your email address and password. It's in the top-right corner of the page. Doing so prompts a drop-down menu. If you had to sign into Skype, skip this step and the next step. This option is near the top of the drop-down menu. It's in the blue column that's on the left side of the page. This will open your profile page. This button is in the upper-right side of the Profile page. Type a new first and/or last name into the appropriately labeled text field(s) near the top of the "Personal information" area. It's a green button in the top-right corner of the page. Doing so will save your name and apply it to your Skype account; the next time you open Skype while connected to the Internet, you should see your new name.
Open the Skype website. Click your name. Click My account. Scroll down and click Edit profile. Click Edit profile. Edit your name. Click Save.