Go to this page : http://people.mozilla.org/~jwein/ux-nightly/ ( or this one ) and download the appropriate build ( 32 or 64-bit, you can find out which one you need by going to Applications > System Settings > About ).
In my case it says : Version: 0.2 "Luna" ( 64-bit ).
Navigate to Downloads and open firefox-27.0a1.en-US.linux-x86_64.tar.bz2 with
Archive Manager ( just click on it ).
You will have a folder named firefox.
Assuming that you have installed and used firefox from the repositories before, I will rename the folder firefox firefox-ux and move the folder to my home directory :
mv firefox/ ../firefox-ux
In the window that will open create a profile : call it ux and start UX with this profile:
If all goes well....
Create a desktop file in ~/.local/share/applications with the following lines :
Exec=/home/username/firefox-ux/firefox -P "ux" --no-remote %u
Replace username with your username.....
The -no-remote switch will make it possible to run firefox with different profiles simultaneously ( see : http://kb.mozillazine.org/Command_line_arguments ).
Log out and in to see Firefox UX occur in Slingshot.