There are many VoIP services available that allow you to send and receive text messages – SMS and MMS – from your PC but this functionality has been missing from Skype for many years. Although you can make phone calls using Skype and do chatting, sending SMS and text messages to normal mobile phones and receiving the same from normal mobile phones cannot yet be done.
In a recent update to Skype Preview for Windows 10 which is only available in the Windows Insider Program you can send and receive text messages from mobile phones. If you are in a hurry, you can go to the Windows Store and download the latest version and check for yourself if you can send and receive text messages from mobile phones or not.
The only problem is that when you try to install the latest Skype version and if some sort of Skype version is already installed on your PC, you are taken to that app instead of allowing you to install the latest version.
If you want to be patient, text messaging will become available in a few months when it is gradually rolled out to all versions of Skype. Yes, this is also a problem with Skype that there are different versions for different platforms.