sudo add-apt-repository ppa:webupd8team/sublime-text-3
sudo apt-get update
sudo apt-get install sublime-text-installer
run /opt/sublime_text/sublime_text
or sudo ln -s /opt/sublime_text/sublime_text /usr/local/bin/subl
with ubuntu bash on Windows 10 and Xming:
DISPLAY=:0 /opt/sublime_text/sublime_text
remove:
sudo apt-get remove sublime-text-installer
file manager:
sudo apt-get install PCManFM
DISPLAY=:0 pcmanfm
It's now possible to launch Windows executables (like Sublime Text) from the Bash on Ubuntu command line - at least, on the Insider Preview (slow ring) version of Windows that I'm using (OS Build 14986.1001). It's fairly easy to sign up for the Insider Preview, and I've had no problems since switching (with either Bash on Ubuntu or Windows generally).You should also be able to set an alias to use it from the command line (as you were trying to do).
Mine is as follows (and works fine):
alias subl='"/mnt/c/Program Files/Sublime Text 3/subl.exe"'
Note the second set of quotes: this is necessary because of the spaces in the file path.