You must also have a compatible version of Google Play Services (typically version 9.x or earlier is recommended for peak stability on Lollipop). Top Sources for Google Play Store APKs
Do not download a "Google Play Store APK for Android 5.1.1 extra." That "extra" usually means "extra malware" or "extra crashes." If your Play Store is broken, wipe the data. If that doesn't work, accept that the device is end-of-life and look into custom ROMs (like LineageOS) or buy a cheap modern phone.
Since you are running an older OS, here are a few extra tips to keep in mind that aren't usually covered in standard guides:
Before you download anything, you must allow your device to install files from outside the official Play Store.