I'm running 11.10 and did a backup with the default program titled "Backup" (Deja Dup).
How can I know that it backed up my files correctly / completely?
as bodhi.zazen explained, the best way would be to restore the backup on a test environment. Then, I suggest you to store somekind of checksum database of your backups after the restoration test
Update Deja-dup to 24.x or latest release.
In later version of 23.x they have added integrity check feature.
Have a look this link. https://bugs.launchpad.net/deja-dup/+bug/1055356