How to Back up phone contacts to google account 

Back up & sync device contacts
Back up & sync device contacts by saving them as Google contacts:
On your Android phone or tablet, open the "Settings" app.
Tap Google and then Settings for Google apps and then Google Contacts sync and then Also sync device contacts and then Automatically back up & sync device contacts.
Turn on Automatically back up & sync device contacts.
Pick the account you’d like your contacts to be saved in. Your contacts can only be automatically saved to one Google Account.