You can create a read-replica of your MySQL 5.6 instance, and then use the upgrade feature provided by RDS to upgrade the read-replica to MySQL 5.7 - the RDS documentation explains how to do this in a fair amount of detail - look for the section titled 'Upgrading a MySQL Database with Reduced Downtime' http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_UpgradeDBInstance.MySQL.html
You can then confirm that you are happy with the new instance, and perform a manual failover to the new instance by promoting it to a primary instance, and either renaming the instances or redirecting your application to the new instance.
You must ensure that you stop your application making writes to the original instance before you promote the new instance to be a primary, and also ensure that the read-replica has processed all outstanding binary log statements received from the original primary instance. It is often best to do this during scheduled application downtime - there are ways of doing this without downtime, but they require much more explanation and testing.
Clearly, you should follow this process in your test environment prior to performing the upgrade on the production environment, allowing you to evaluate the compatibility of your application with the new version of MySQL.