By far, the simplest way to update CoreELEC would be to enable the Auto Update feature.
To do so, start the CoreELEC Configuration addon and, to the right hand side of the first screen, you will see the Updates section.
Simply ensure that the word auto is displayed to the right of the Automatic Updates option and you will always be kept up-to-date with the latest builds.
In addition, there are two other different ways to access (and therefore update) the box you installed CoreELEC on; via your preferred file explorer using Samba shares or using SSH.
Whichever method you choose must have been enabled in the CoreELEC Configuration addon, prior to being available.
For manually updating, first check CoreELEC latest releases for the newest updates.
Updating using the Samba shares
For updating CoreELEC using Samba shares, you have to download the correct .tar file for your actual hardware, and copy it to the update folder on your CoreELEC device.
Using a file browser, access the Samba shares following this guide (not forgetting to replace references to LibreELEC with CoreELEC) and, after copying the file to the update folder, reboot the CoreELEC device.
The update process will start after that by unpacking and installing the update, followed by a another reboot.
Updating using the CLI
For updating CoreELEC while using the command line interface, you have to SSH in, using username: root and password: coreelec and then move to the correct folder with the command cd /storage/.update
Once at the correct folder, the specific file can be downloaded directly to our device using
wget https://github.com/CoreELEC/CoreELEC/releases/download/8.90.5/xxxxxx.tar
(substituting 8.90.5 with latest version number and xxxxxx.tar with the correct file for your device).
If preferred, a previously downloaded update file can, of course, be transferred to the /storage/.update folder, using your favourite SSH software.
Once the update file is in the correct folder, reboot the device
The update process will start after that by unpacking and installing the update, followed by a another reboot.
Adapted from source, credits to original authors.