EDIT: January 16, 2007
This thread is officially obsolete as flash 9 is finished and out of beta. But, since you came here looking for a way to install flash 9, I will just put the instructions for the final version.
Note, I have done this on edgy. If anyone wants to confirm that it works on any other versions, feel free.
Open the terminal, Applications>Accessories>Terminal, I will assume this stuff will all being happening in the home directory as that is where the terminal opens by default.
First remove the previously installed flash plugins (obviously if you don't have any installed, disregard).
There are a few ways to do it depending on the way you installed it. If you are not sure, you can do all of them without damaging anything but if you do all of them, make sure you do them in order. If you know how you installed the old one initially, choose the appropriate one.
1. Removing it via apt
2. Removing it manually if it is in the /usr/lib/firefox/plugins directory
sudo apt-get remove flashplugin-nonfree
If it tells you that a file doesn't exist, don't worry about it. Flash 7 had flashplayer.xpt but flash 9 beta 1 didn't, so depending on which version you had installed, you may get an error that flashplayer.xpt doesn't exist. Or if you don't have any of them installed, they both won't exist. Either way, just carry on.
sudo rm /usr/lib/firefox/plugins/libflashplayer.so /usr/lib/firefox/plugins/flashplayer.xpt
3. Removing it manually if it is in the ~/.mozilla/plugins directory
Same thing applies as number 2 as far as errors are concerned, just disregard.
rm ~/.mozilla/plugins/libflashplayer.so ~/.mozilla/plugins/flashplayer.xpt
Second get the tar.gz file
Third extract the tar.gz
Fourth change into the new directory
tar xf install_flash_player_9_linux.tar.gz
Finally run the installer script and follow the prompts
When it asks for your mozilla, seamonkey, or firefox directory, tell it /usr/lib/firefox
That should be it. If you want to remove all the excess garbage that you downloaded, just change back into the home directory and remove it.
rm -rf install_flash_player_9_linux*