Open up an (elevated) Powershell instance and run the following script. It will download Ubutu and enable the Linux Subsystem on Windows.

# download Ubuntu
Invoke-WebRequest -Uri https://aka.ms/wsl-ubuntu-1604 -OutFile Ubuntu.zip -UseBasicParsing

# extract the file
Expand-Archive ./Ubuntu.zip ./Ubuntu

# set path
$userenv = [System.Environment]::GetEnvironmentVariable("Path", "User")
[System.Environment]::SetEnvironmentVariable("PATH", $userenv + $(Get-Location).Path + "\Ubuntu", "User")

# install WSL:
Enable-WindowsOptionalFeature -Online -FeatureName Microsoft-Windows-Subsystem-Linux

Now that everything is in place, please restart the server. After the restart, open up Powershell again and run Ubuntu. You will be asked to create a new UNIX user.

cd Ubuntu
ubuntu1604.exe

Now you can upgrade to the latest version of Ubuntu by this script:

sudo apt-get update
sudo apt-get upgrade

It might take some time to upgrade.

Improvements
2019-06-24: Changed the out file for the download link into a zip extension, no renaming needed anymore.
2018-06-24: The executable in the current download is ubuntu1604.exe, so I changed it in the script.