Depending on your OS, you may have to issue one of the following command:
$ yum update pydio
or
$ apt-get install --only-upgrade pydio
Let the package manager do its job.
After the packages upgrades, please have a look inside the /var/lib/doc/pydio/upgrade/db folder. This folder may contain a set of scripts named with version numbers for upgrading from one version to another. If you find that one or many scripts are required to go from your previous version to the new one, you will have to apply them all. For example, if you are running v6.0.5 and just upgraded to v6.0.8, you may have to run the scripts 6.0.5-6.0.6, 6.0.6-6.0.7 and 6.0.7-6.0.8.
Scripts are also suffixed with the according DB extension: .mysql is for MySql / MariaDB, .pgsql for Postgres and .sqlite for SQlite. Make sur to use the correct one.
Once you have collected the necessary scripts, use either a GUI-based tool to connect to your DB and exectute the SQL queries (like PhpMyAdmin for example)
or apply the SQL by using command line.
For example for MySql:
$ mysql -u adminuser -p DBNAME << pydio-core-upgrade-6.0.7-6.0.8.mysql