How to troubleshoot Time Machine when backup always halts at some point?
Troubleshooting this is fairly straightforward to start.

When it hangs, open up Applications/Utilities and launch terminal.

Type:

 sudo tmdiagnose


That will gather many logs and statistics and allow you to have a complete picture of what Time Machine was stuck on. After collecting three such instances (reboot the Mac after the script ends), you can start looking for patterns or a specific file that's causing things to hang.

You also can capture very precise details before and after the tmdiagnose command by issuing a tmutil status command.

I've been troubleshooting this on one Mac for a few weeks now, so it's not always clear what or where the issue lies. Hopefully yours will be easier to nail down than mine.