First, I’m assuming you’ve got both an EC2 instance and an RDS instance in the same VPC, as mentioned in my article on setting up a Rails application to deploy to Elastic Beanstalk. I’m also assuming you’ve created a .pemkey and downloaded it to connect to this EC2 instance

configure an alias in ssh config to connect to the EC2 instance

Host my_app
  HostName xx.xx.xxx.xxx
  User ec2-user
  IdentityFile ~/.ssh/my_app.pem
  ForwardAgent yes

test the ssh config: ssh my_app

Last login: 
 _____ _           _   _      ____                       _        _ _
| ____| | __ _ ___| |_(_) ___| __ )  ___  __ _ _ __  ___| |_ __ _| | | __
|  _| | |/ _` / __| __| |/ __|  _ \ / _ \/ _` | '_ \/ __| __/ _` | | |/ /
| |___| | (_| \__ \ |_| | (__| |_) |  __/ (_| | | | \__ \ || (_| | |   <
|_____|_|\__,_|___/\__|_|\___|____/ \___|\__,_|_| |_|___/\__\__,_|_|_|\_\
                                       Amazon Linux AMI

This EC2 instance is managed by AWS Elastic Beanstalk. Changes made via SSH 
WILL BE LOST if the instance is replaced by auto-scaling. For more information 
on customizing your Elastic Beanstalk environment, see our documentation here: 
http://docs.aws.amazon.com/elasticbeanstalk/latest/dg/customize-containers-ec2.html
[ec2-user@xx-xx-xxx-xxx ~]$

add this alias to ~/.bashrc

alias my_rds='ssh -N -L 55432:my-app-db-instance.us-east-1.rds.amazonaws.com:5432 my_app'

you’re using your my_app ssh config to forward traffic from my-app-db-instance, port 5432 to localhost 55432

Traffic from your db instance will now forward to 55432, you could theoretically change your .envfile to have this configuration

database_name: 'my-rds-db-name'
database_username: 'my-rds-db-username'
database_password: 'my-rds-db-password'
database_port: '55432'
database_url: 'localhost'

and your config/database.ymlto have this configuration

development:
  <<: *default
  database: <%= ENV['database_name'] %>
  pool: 5
  username: <%= ENV['database_username'] %>
  password: <%= ENV['database_password'] %>
  host: <%= ENV['database_url'] %>
  port: <%= ENV['database_port'] %>
  reaping_frequency: 10

As a note of warning, if you’re doing this with a production RDS instance while doing local development, you’re making changes to live production data while doing dev work, definitely not recommended.