To create a jenkins slave with firefox, a $5 a month droplet from Digital Ocean is more than enough to run automated tests. If you need to set up a Jenkins master, feel free to check out my instructions in github.
For this setup, I used Ubuntu 12.04.4 x64 as my linux kernel. Once the droplet is created, ssh into the box and follow the instructions below.
Installing Firefox on Ubuntu
$ apt-get update
$ apt-get install xvfb firefox imagemagick
Additional fonts to install for xvfb
$ apt-get install xfonts-100dpi xfonts-75dpi xfonts-scalable xfonts-cyrillic
$ apt-get install msttcorefonts cabextract x-ttcidfont-conf
Now, let's verify the installation.
$ Xvfb :1 -screen 0 1280x960x24 &
Open Firefox and visit the homepage
$ DISPLAY=:1 firefox http://www.sqaessentials.com &
Take a screenshot
$ DISPLAY=:1 import -window root -crop 1264x948+0+0 -resize 200x150 -quality 90 ~/screenshot.jpg
Download the screenshot and you should see the browser open to www.sqaessentials.com.
The machine is now ready to add to your Jenkins infrastructure.