Why does SQL Server Management Studio hang (Not Responding) when connecting to Amazon RDS?
Sounds like the connection is successful but SSMS is having trouble loading metadata to populate Object Explorer. I would try opening SSMS without connecting to this server, open an empty query window, disable IntelliSense, and then change that query window's connection.

Also your copy of SSMS is quite old, I would strongly consider applying SP3 + CU1 to your workstation, or moving up to 2014 SP1 or even the new monthly previews. I don't know what build RDS is on, but you should always strive to have SSMS >= the highest version of server you have to manage.