The instructions can be a bit confusing. Basically you just go down to your current version in the UPGRADING.html file, and then work your way up, running each altertable that there is a link for. Once you get to version 6.4.1, then there is a new method of upgrading with the install/upgrade.php script from there.
As you work your way up, it will give you information on exactly what you need to do until you get to 6.5.5.
