MySQL DB Migration from on-premises to AWS through replication
Here's pretty detailed blog post about how to do this:
How to Migrate to MySQL RDS Without Downtime

Our approach was to configure M/M replication between our MySQL instance hosted on EC2 and the new RDS, then gracefully switch traffic from our application servers one at a time without any downtime for our customers.
To clarify, the Non-RDS server in this article refers to MySQL instance running either in your data center or on an Amazon EC2 instance