I will answer this question with Agile iterations in mind. (An iteration could be a full Sprint or a single-user-story-cycle if you do Kanban or swarming.)
The QA department is understaffed when:
- Not every cross-functional Agile team has a someone with QA knowledge
- The ratio defects versus released features is getting out of control
- Test automation is not completed during the iteration
- Manual testing is executed in the next iteration
- Working software is not shipped at the end of iteration (This could have other causes as well.)
- Un-tested software is shipped
- QA employees have no time for a QA-community which focuses on growing and learning from each other during work hours
we've built a rather long backlog of test stories and
scenarios to be automated
I have learned what helps best is to visualize it. Somewhere prominent in your company show you are behind constantly. Just make a QA metrics area in the main hallway. Post things as test-automation cycle-time and lead-time, number of stories still to automate, defects vs new features. Preferable add some targets you would like to achieve with a trend arrow per day/week/month.
I am slightly worried by your words "long backlog", this clearly shows the work is not done. Accumulating work not done should be alarming to management. I would push the development teams to add testing (including test automation) to their definition of done. Adding more test-engineers will probably not solve your problems anyways.
Experiment with scaling test-engineers by making them part of the development teams, let the team figure out what their perfect ratio devs versus testers should be. This will be different per team. But integrating testers in the team has a lot of benefits. Whole team quality attitude, better knowledge sharing, no-handovers and really DONE!DONE! (e.g. production ready) to name a few.