The Google Play Store may have fewer restrictions that its iOS counterpart, but that doesn't mean that all apps make the cut. Ad blockers and Xposed modules that bypass paywalls have never had a place on the Google Play Store, which makes tracking their app-updates a painful process.
This struggle of keeping non-Play Store apps up to date led XDA user JusticeRage to create ApkTrack, a lightweight tool that checks all of your installed apps for updates. This does mean that the app will check your Play Store apps as well, so keep that in mind.
ApkTrack is not hosted on the Google Play Store. However, you can easily download it directly from the developer. After the download completes, open the APK file and install the app. You should also be sure to have installation of apps from "Unknown sources" enabled as well.
When you first open ApkTrack, you will need to tap the Search icon (a magnifying glass in a box) on the top right. This will scan your device for new apps that were installed since the last time the app was used. By default, it will run a scan when it is opened for the first time. After the app list has been populated, you can tap the Sync button (just right of the Search icon) to check for updates.
After the update check has completed, you will receive a notification letting you know if any apps need to be updated. Also, a Search icon (just a magnifying glass) will appear next to each app that has newer versions available. Tap on that to use the app's built-in search engine to find an available APK to download. ApkTrack does not handle the downloading or installing of updates, so you will have to do that yourself.
If you'd like to have ApkTrack check for updates periodically in the background, select Settings from the overflow menu and check the box next to "Enable background checks." There you will also find an option to select which search engine to use when searching for updates to your APKs.
Did you find ApkTrack useful for updating your non-Play Store apps?