Now you need to logout and login again or restart your computer.
Click the IM icon in menubar and Select Text Entry Setting
In Text Entry Setting window click + icon (marked by arrow) in left pane. Write avro in the search box (marked by arrow). Choose Bengali (Avro Phonetic) from the list and press Add button to add Avro to the list.
(Optional) The keyboard shortcut to switch input method has been changed to Super+Space since Ubuntu 13.10. You can change it to your preference from Text Entry Setting window.
Open any text editing software (like, gedit). Now Press Super+Space to toggle between English and Avro (Bengali)
Avro Phonetic has now been made available to Ubuntu users via an official package.
To install it:
sudo apt install ibus-avro
ibus-avro in the Ubuntu archive installs the latest upstream software, including a bugfix which is important to Ubuntu users. The GitHub repo referred to in step 1 and 2 below is in read-only mode, and not useful any longer.