Copying the entire drive is a really bad idea for several reasons. Copying it to the cloud is also a bad idea. Buying a new TB or larger drive is far cheaper, faster, and easier and would let you just start backing up and put the "corrupt" backup on the shelf until you're sure you don't need to recover any files from it.
Since Time Machine can back up to more than one destination, I would add a second drive to have a temporary or permanent second copy of backup data.
Due to the hard links that are used to save space when a file isn't changed, your backup method will have to deal with these links to avoid increasing the data transfer and storage requirements (depending on how many backup intervals are stored - you could end up with Petabytes of data from a 500 GB store of data).
It's much, much better to just pick one or two intervals to back up or let a tool like BackupLoupe interpret for you which files changed at which point in the backup so you can intelligently copy the files you really need off the drive.
Again, if you have more than 1 GB of files, I would avoid DropBox and slower clouds and get a tool crafted for large transfers like Arq or Transmit and push your files to Amazon S3 or possibly Glacier if you have multiple terabytes or more to store but won't likely restore any or all of it once you've determined what's really of value in the backup drive.
In practice, anything that lets you store unlimited data in the cloud will take far too long to get there, so minimize what you send up or pay for reasonable speed for the time you'll need it in the cloud.
That being said, if you really need to store it all, use Disk Utility to create an image of the drive (which hopefully can be done - it might fail to comply depending on the corruption that is present) so that you only copy the actual space used and don't get the hard links expanded into multiple copies of each file. Once you have a dmg (or compressed dmg), you can upload that.