If you cannot sign in to your Google Account after installing the APK, the device's "Google Play Services" app is likely missing or corrupted.
If you must use Android 4.2.2, download the Play Store APK from a trusted mirror like APKMirror – search for "Google Play Store 16.8.23". Also install Google Play Services 11.9.75 (latest for Android 4.2.2). Be aware that many apps will no longer work. play store apk android 4.2.2