How to speed up 21.5inch, late 2013 iMac processor

Get an external usb3 ssd to use instead of the dog slow 5400rpm internal hard drive.

Welcome to the Apple Support Communities.

While SeaPapp is almost certainly correct, I always like to verify this before suggesting something to purchase. For this reason, I believe we need to proceed with an EtreCheck report. EtreCheck is a useful App to diagnose and identify issues, both hardware and software. It is trusted by many users. Please follow the steps outlined below:

1. Navigate to: https://etrecheck.com and get the free version of EtreCheckPro.
2. Download it. It is a free and trusted app. It is often recommended here on Apple Support Communities. 
3. Open EtreCheck, and before running the report, be sure to check “Allow full Drive Access” at the bottom of the EtreCheckPro window.
4. Run the report, and put it in a post, as shown here: https://discussions.apple.com/docs/DOC-250000211. We will then examine the report for issues in your Mac.

Cheers,

Jack