Disclaimer: This article is for educational purposes. Android 5.1 is no longer supported by Google for security patches. Use manual APKs at your own risk, and consider upgrading your hardware when possible.
If you have tried three different versions of the Play Store APK and still have problems, consider these alternatives:
Before you can install an APK on Android 5.1, you must give your device permission to install apps from sources other than the Google Play Store.
LineageOS 14.1 or 15.1 can bring Android 7.1 or 8.1 to some Lollipop-era devices. This updates the entire system, including the Play Store framework.