Migrate heroku database to Amazon RDS with minimum downtime
Pretty sure you are going to have to perform a good old-fashioned dump-and-restore here. Amazon doesn't seem to offer any other import options than loading in a dump:

http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/PostgreSQL.Procedural.Importing.html

and even if they did, AFAICT the only export options offered by Heroku are pg_dumps (i.e. not basebackups or the ability to set up an external hot standby node):

https://devcenter.heroku.com/articles/heroku-postgres-import-export

If you're lucky, you're using a Postgres version new enough to support both parallelized pg_dump (9.3+) and parallelized pg_restore (8.4+), which will make your job a bit quicker. And hopefully you can afford to take a maintenance window during which you lock out writes to your site while the switchover is underway...