Time Machine: Hangs on "Indexing" message

My AirPort Time Capsule isn't working, and my iMac Hard Drive is failing, so I'm trying to do a full Backup before taking it in to Apple for a replacement Hard Drive (and Fan).

As my Time Capsule isn't working, I'm trying to back up to a SeaGate 2TB Backup Plus (which isn't faulty as I've been able to save iPhoto Library files to it). My Macintosh HD has used 495GB out of 1TB.

After initiating the Backup via Time Machine, the pop-ups I get are:

"Calculating size ..."

and then

"Indexing backup …"

And I can't seem to get past the "Indexing backup" message. I stopped the Time Machine Backup after it had been running for 24 hours.

In Finder, if I click on the Drive that I'm saving to, there is a folder with the following in it:

Backups.backupdb > [MY_MAC] > 2016-02-13-141047.inProgress

Looking for some assistance on how to get past the Indexing issue.

Many thanks.