In order to create a custom Skype ID, you should have a custom e-mail address.
For example if you have an e-mail address like "firstname.lastname@example.org" the chances of you having a Skype ID like live:jane123456 is very high. But if you create a more specific e-mail address like "email@example.com" the chances of you having a Skype ID like live:jane.brunette is high.
To do this, you have two options. You can choose which one that fits best for you.
You go to www.skype.com, click on Sign in and then select New to Skype (as in the image below).
Then select Use email instead
a) You create a new e-mail address before signing up on Skype and fill it in here
b) You select Get a new email address
You create your new @outlook.com address
Add your first and last name that will show up whenever someone will search yo on Skype
And then you're all set.
You will be automatically logged into Skype for web (make sure that you don't try to use the SkyPrivate plugin on Skype web since it won't work).
In order to find out your new Skype ID, click on your name and it will be available.