Recently I had some problem with signing in to Skype on Windows10. Here is a way to solve the problem, it worked for me.
Basically you make sure first Skype is not running (exit app, remove from Start apps list, restart PC). Then this tutorial will guide you how to disable or delete your user cache saved in Skype app folder. There is some bug with not deleting some lock file, so you get rid of it this way. Then Skype starts clean from scratch. More on thi link: