I meet the same problem with you and I follow these steps.
step 1. Go to Settings > Apps (or Applications) > manage applications and select show All. Then find Google Services Framework and click the Clear Data button. Then click the Force Stop button. then uninstall GoogleServicesFramework.
step 2. Now install successfully Google play services.apk file
step 3. then install GoogleServicesFramework.apk file n restart bluestacks.
NOTE: Bluestacks Restart required. Go to taskbar right click on bluestacks and Quit.
step 4. start bluestacks n again sign in your gmail account. That's it! Enjoy!
P.S : You should Google some APK files Download links.