You need to get your contacts saved to your Google account, if they are not already. Log into your Google account online and check to see if all your contacts are in you Gmail contacts list. If they are you're good to go. To make sure they're saved to Google in the future, just go to:
Settings > Accounts > Google > Select your account > Enable contacts sync.
This is enabled by default.
If your contacts are not in Gmail, the following steps will put them there. You have saved contacts to local storage on the phone only. To export those contacts, import them to Gmail, delete all contacts on the phone and resync clean a contacts list to the phone follow these steps:
1. Turn off sync in the phone. This is to prevent it from attempting to sync while changes are made.
2. Open the Contacts app and tap Menu. Select Import/export, then Export to storage. The contacts will be exported as a .vcf file. Make a duplicate copy of the .vcf as a safety backup.
3. Delete all contacts on the phone. Do:
Settings >Apps > Google contacts > Clear cache > Clear data.
When finished, there should be no contacts on the phone.
4. Login to Gmail and import the .vcf file into Gmail contacts.
5. In Gmail contacts, click on More and select Merge duplicate contacts.
6. Check the list for errors and do any needed edits. When satisfied export a .vcf file copy for backup.
7. Enable contacts sync on the phone and wait for the corrected contacts list to download.
Done.