What's the best way to backup a system encrypted with whole disk-encryption? I mean, The total system is encrypted, not only the home folder.
If you have whole-disk encryption, then just boot the system in the usual way, including unlocking the disk. Then make a backup using say duplicity, using gpg so that it's encrypted on the destination machine.
There is not much point trying to backup the raw encrypted partition.
If, when you say
What happens with the encryption placed on your home directory is that every file and folder in it is copied to a encrypted directory that mirrors your home directory ie you would have
where the second directory is the encrypted one. When you log in, the the security system attempt to decrypt the encrypted directory using your username and password, thus giving you access to the stuff. It's not encrypted while you're logged in, so you can back the stuff up normally.
You can read up on how the encrypted home directory works at this Ubuntu Wiki page. Specifically, you should check out the How it works section mid way down the page, and excerpt of which I've posted below: