My HDD crashed and am now using the LiveCD I ordered from shipit.ubuntu.com, which allows me to download onto the disc. I downloaded the tar.bz2 file and extracted, now what?

Which Linux distribution are you using? Most distros have Firefox in their repository, for example in a .deb based distro you can "apt-get install firefox" or in an .rpm distro "yum install firefox" etc.

If for some reason your distro does not have a package manager, then I can tell you the download from Mozilla does not need to be "installed." Simply extract it with your favorite archive manager, then double-click the Firefox executable. (It is just like an .exe file in Windows.) You can put a shortcut on your desktop if you like.

I'm using Ubuntu 8.04, it has Firefox 3.5, but I don't know how to upgrade.

Am I supposed to update at the repository? If so, how do I get there?

To upgrade your Ubuntu system, you can use the graphical Update Manager or the following terminal commands:

According to the Ubuntu website, this should give you Firefox 3.6.13:
http://packages.ubuntu.com/hardy/firefox

Ubuntu 8.04 was released April 2008, so in my opinion it is very important to update your system to get all the security patches and bug fixes from the last 2.5 years!

Please open a terminal window, and type : sudo apt-get update

And then : sudo apt-get install firefox-3.0 firefox
This will provide firefox version 3.6.13 .

Synaptic can be used for update / package install too :
Update : the 'Refresh' button.

..

Thanks knudfl, Happy New Year!

Running a Live CD is a very temporary solution, as all changes you make (applications installed, documents created, configurations changed, etc.) will be lost when you power down the computer. You also are limited by the amount of RAM you have.

That being said, Ubuntu is Ubuntu, whether you are running from the Live CD or from a hard drive, and the commands that knudfl and I suggested should work equally well.