Go to
https://www.skype.com/en/
At the top of the page, on the right hand side, click on Sign in. A box will pop up with three options - My Account, Use Skype online and New to Skype ? Sign up.
If you already have a Skype id or a Hotmail or Live mail account, choose the Use Skype online option which will take you to the log in page. If you don't have any of those, click on the Sign up bit to create an account.
When you sign in the site does say that Skype online is in Beta, so there may be some problems with it but it does seem to be usable (I have one friend who accesses Skype that way and hasn't had any problems).