You can read my guide in installing apps in your home folder here
as for me , I install firefox in ~/local/opt/firefox2 and a symlink to the firefox startup script in ~/local/bin
Btw, if you are using a compositing manager (eg. compiz/beryl ), you might need to add these two lines into the firefox startup script - right after the GPL notice - to keep firefox from crashing when opening sites with Adobe Flash
XLIB_SKIP_ARGB_VISUALS=1
export XLIB_SKIP_ARGB_VISUALS
G'luck and enjoy~ :)