The One with the Thoughts of Frans

With a bit of trial and error it worked out like this. Alternatively you could use /opt.

Create a file called something like firefox.desktop in /home/username/.local/share/applications/

In that file, put:

[Desktop Entry]
Type=Application
Name=Firefox
GenericName=Web Browser
X-GNOME-FullName=Firefox Web Browser
Exec=/home/username/programs/firefox/firefox %u
Terminal=false
X-MultipleArgs=false
Type=Application
Icon=/home/username/programs/firefox/browser/icons/mozicon128.png
Categories=Network;Webbrowser;
MimeType=text/html;text/xml;application/xhtml+xml;application/xml;application/vnd.mozilla.xul+xml;application/rss+xml;application/rdf+xml;image/gif;image/jpeg;image/png;x-scheme-handler/http;x-scheme-handler/https;
StartupWMClass=Firefox-bin
StartupNotify=true

To make it the default system-wide, manually add it to Xfce’s Preferred applications or your window manager’s equivalent. Also add it to Debian’s alternative system.

$ sudo update-alternatives --install /usr/bin/x-www-browser x-www-browser /home/username/programs/firefox/firefox 20
[sudo] password for username: 
$ sudo update-alternatives --config x-www-browser
There are 2 choices for the alternative x-www-browser (providing /usr/bin/x-www-browser).

  Selection    Path                                   Priority   Status
------------------------------------------------------------
* 0            /usr/bin/iceweasel                      70        auto mode
  1            /home/monkey/programs/firefox/firefox   20        manual mode
  2            /usr/bin/iceweasel                      70        manual mode

Press enter to keep the current choice[*], or type selection number: 1
update-alternatives: using /home/monkey/programs/firefox/firefox to provide /usr/bin/x-www-browser (x-www-browser) in manual mode

It’s not very pretty, but it does the trick.