Here is the complete process of backing up your daily WHM backup file(s) to Dropbox.

Pre-requisites
a. Enable backups in WHM: Home/Backup/Configure Backup/Backup Status:Enabled
b. In case you don’t already have one yet, create a Dropbox account (referral link)

Once you have these pre-requisites, you can perform the following operations:
1. Create a Dropbox app dropbox.com/developers/apps
2. Download the Core APIs SDK. I downloaded the PHP flavour and put it in /home/user/dropbox/php-sdk-1.0.1
3. Make sure that the php_curl extension is enabled
4. Create /home/user/dropbox/php-sdk-1.0.1/backup – which will hold the necessary files
5. In the above directory, create a file called login.json with the following content:

{
  "key": "INSERT_APP_KEY",
  "secret": "INSERT_SECRET",
  "access_type": "FullDropbox" or "AppFolder"
}

6. Copy the files ‘authorize.php‘, ‘helper.php‘, ‘upload-file.php‘ from examples/ to backup/ directory
7. Authorize your Dropbox app:

php authorize.php login.json auth.json

This will read the credentials in login.json and attempt to create auth.json file. It will prompt you to open a Dropbox URL where you will authorize your Dropbox app to use your Dropbox account. After you allow, go back to shell, press enter and the auth.json file will be created
8. You’re ready to go. Assuming that WHM creates your daily backups in /backup/cpbackup/daily/ folder, you can run the command below to transfer your tar.gz backup file to Dropbox.

php /home/user/dropbox/php-sdk-1.0.1/backup/upload-file.php /home/user/dropbox/php-sdk-1.0.1/backup/auth.json /backup/cpbackup/daily/user.tar.gz /user.`date +\%Y-\%m-\%d`.tar.gz > /home/user/dropbox/logs/user.`date +\%Y-\%m-\%d`.log

This command will save the output of the operation to a log file for further reference.

9. Email [optional] You can change the command above to send the output by email:

{command above} && mail -s 'Daily backup' 'yourATemail.com' < /home/user/dropbox/logs/user.`date +\%Y-\%m-\%d`.log

10. Cron Put the command at step 8 (or 9) above in a cron: crontab -l or edit the file /var/spool/cron/root and make the changes directly.
Check the new cron jobs: crontab -l
Check the cron log: tail -f /var/log/cron