To become partner of record you will need your Partner ID. You can find this in the partner portal from microsoft: https://partners.microsoft.com/
When you have your Partner ID you have to log in with the customers administrator office365 account on portal.office.com.
Go to -> Billing -> subscriptions.
Select the subscription on the right, at the actions panel choose, Add partner of record.
Copy your Partner ID and click Ok.
You are now a partner of record for this customer.