[stumped]

Copy, don't move, the entire calibre library folder from your pc to a folder of the same name in Dropbox.
You need to have the Dropbox app installed on your pc then you copy it into the local Dropbox folder and leave it to sync.
Optionally you can separately copy the calibre config library from the hidden folder in app data. To a folder in dropbox. Call it calibre config or some meaningful name

Use the Dropbox app because if you try to upload to dropbox.com via a browser, it can mess up the subfolder structure that calibre needs

Once it is done you can thereafter use a file sync utility to mirror any changes to Dropbox. I use free file sync but there are others. Set it to do a one way sync to keep the Dropbox copy as a mirror of the source library. Run it whenever you make changes and it will only take a minute to reflect those into dropbox

So my source is is the folder calibre library in my documents. My copy is to a folder called calibre library in my dropbox
And Google free file sync for the freeware installer, which no longer includes any crapware. It just asks for an optional donation.

Checkout an app called calibre box in Google play store if you want to view and download books from Dropbox to a phone or twblet