This won't be upgraded with "apt-get upgrade"
This is being run on a very sparse system. Here are the packages you should probably get, if you don't already have them:
apt-get update && apt-get install xorg iceweasel bzip2 libdbus-glib-1-2
Download the most recent version of Firefox. You can just go to http://getfirefox.com with Iceweasel (or any other browser), or you can use wget from the command line:
Now just unpack the file where it belongs. I'm not actually sure what the best place for it is, so I put it in /usr/local/src. You'll need to have root privileges for this, and you'll need to note the location of the file you just downloaded (in my case it's /home/james/firefox-3.6.3.tar.bz2).
su cd /usr/local/src tar -xvjf /home/james/firefox-3.6.3.tar.bz2
Now you'll have a directory named "firefox" where you just unpacked it -- /usr/local/src in the example above. Now you just need to symlink it /usr/local/bin (you should still be root).
ln -s /usr/local/src/firefox/firefox /usr/local/bin
That's it. There's no need to get rid of Iceweasel. If you really want you can create separate profiles for Iceweasel and Firefox. By default they'll use the same one. Launch it and browse (as a regular user, not as root):