How to install Play Store on Alcatel OT 979
You can't just install the APK, you have to download a GApps package, which contains all the Google applications that come with Google licensed phones. This, however, requires rooting the phone and installing a recovery.

The reason for all the trouble is that while Android is open source, Google Apps are proprietary code, and devices have to be licensed with Google in order for the Apps to be included on the device.

For more information, see How can I install the Google Apps Package (Play Store, ...) on my Android device?, RootzWiki GApps page and GApps download.