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.

Summary:
Open the Skype website. Click your name. Click My account. Scroll down and click Edit profile. Click Edit profile. Edit your name. Click Save.