How can I keep copies of my iCloud documents in Dropbox?
I am not absolutely certain, but I believe this is impossible.

For one thing, there isn't iCloud support for simply syncing files. Apps are expected to keep their synced files in their own folders (and are restricted to accessing this folder alone). This means that there's no way to have an iCloud document accessible my multiple apps.

For another thing, even if you were willing to concede and only sync, say, Pages documents with iCloud and Dropbox (hypothetically), Pages for iOS documents are very different from Pages for Mac documents. Here are two screenshots, one of the inside of a Mac Pages document and one of an iOS Pages document, respectively:

 

Post script: hypothetically, though, if you could figure something out, it would probably involve somehow linking ~/Library/Mobile Documents with Dropbox.