First you will need a Microsoft account (Outlook or live or Hotmail account) which might or might not be connected to your skype account.

Log into your Outlook account to see Messaging option as shown below

Clicking on Messaging will open the following window where you can search your contacts and make your first skype web call

After selecting your contact the chat window will show a Call option and a Video Call option. If you are calling through web first time it will ask you to install skype web plugin before you can make call like in facebook video call otherwise call will start but yes the person in call should also be online

Just click on Agree to continue installing the plugin…

Download has begun for the Skype Web plugin after which you will need to install it

Double-click on the downloaded plugin and wait for the installation to complete.

Download and installation takes seconds and you will soon be on a call,Just click Start Call to make your first call

Enjoy Calling!!