First of all you have to activate the Windows Subsystem for Linux from Turn Windows features on and off, which you can find with a simple search from the start menu.
Download, install and start Xming, our substitute for the X server usually found on linux. This component will render the windows of the GUI program we'll throw at it.
At this point you have to install a program with a graphic interface. For this article I'll use a text editor named gedit, but you can use pretty much everything that comes to your mind. I was even able to succesfully run xfce4 straight from the default repository.
sudo apt-get install gedit
After the installation is completed, we "link" the shell to Xming by executing
If you want to avoid having to write the same command every time you restart bash, you can just append it to .bashrc, found in your home directory.
Now you just jave to run the program by writing its name, like this:
You can make shortcuts to start GUI Linux programs straight from your desktop, using the "bash -c" command.
The command of the shortcut is:
bash -c "export DISPLAY=:0; [[[PROGRAM NAME]]]"
Since bash -c doesn't run the content of the user's .bashrc it needs the first line to specify the display before every other command is executed. Of course you have to make sure Xming is running every time you try to open these shortcuts.