See if this guide helps you.

Installation

There are currently three ways to do this, manually, PPA or Ubuntu Make:

Manually

Download from here: Firefox Developer Edition — Mozilla

Extract it with file-roller and move the folder to its final location. A good practice is to install it in /opt/ or /usr/local/.

Once you moved the files to their final location (say /opt/firefox_dev/), you can create the following file ~/.local/share/applications/firefox_dev.desktop to get a launcher with an icon distinct from normal Firefox.

[Desktop Entry] Name=Firefox Developer GenericName=Firefox Developer Edition Exec=/opt/firefox_dev/firefox Terminal=false Icon=/opt/firefox_dev/browser/icons/mozicon128.png Type=Application Categories=Application;Network;X-Developer; Comment=Firefox Developer Edition Web Browser.

To launch it, search for Firefox Developer, and simply run the firefox binary, and voilà.

PPA

Firefox for developers currently resides in Firefox Aurora builds : “Ubuntu Mozilla Daily Build Team” team. It should be noted that:

Firefox Developer Edition replaces the Aurora channel in the Firefox Release Process. Like Aurora, features will land in the Developer Edition every six weeks, after they have stabilized in Nightly builds. - Developer Edition - Mozilla | MDN #The_latest_Firefox_features

Also:

Version 35.0a2, first offered to Firefox Developer Edition users on November 10, 2014 - Firefox — Aurora Notes (35.0a2) — Mozilla#main-content

Install the Firefox Developer Edition, formerly know as Aurora, like so:

sudo add-apt-repository ppa:ubuntu-mozilla-daily/firefox-aurora
sudo apt-get update
sudo apt-get install firefox

Due note that installing with the aforementioned ppa, will result in your current Firefox installation being replaced!

Ubuntu Make This method has been provide by Jorge Castro, I am adding this here, because its also an excellent way to install the Firefox Developer Edition(and many others dev tools).

 sudo add-apt-repository ppa:ubuntu-desktop/ubuntu-make 
 sudo apt-get update
 sudo apt-get install ubuntu-make

After you have installed ubuntu-make, tell it to install the web tool firefox-dev.

 umake web firefox-dev

Source: http://askubuntu.com/questions/548003/how-do-i-install-the-firefox-developer-edition

Using the tarballs will be the easiest way to have multiple builds on Linux. Just extract them in own folders (say in /home to make permissions easier for updates) and make sure each is using a separate Profile is the basics as you should not share a Profile.

https://support.mozilla.org/en-US/kb/profile-manager-create-and-remove-firefox-profiles

Note that Beta builds gets say six to twelve builds for a version. Aurora (aka developer edition) and Nightly gets checkins almost everyday so they get updates each of those days. This means there can sometimes be issues until fixed, finished or reverted.