I\'ve recently gone through the database migration process as outlined here:
https://devcenter.heroku.com/articles/migrating-from-shared-database-to-heroku-postgres<
Steps based on Ashton's answer to upgrade from Dev (10k rows limit) to Basic (10M rows limit)
check db rows exceded limit
heroku pg:info
disable the app and workers to ensure no db changes during db upgrade
heroku maintenance:on
heroku ps:scale worker=0
if you don't have pgbackups
heroku addons:add pgbackups
backup database and get backup id
heroku pg:backups:capture
add db with web interface
view heroku config, you should see new db URL
heroku config --remote heroku
restore backup to new db
heroku pg:backups:restore BACKUP_ID NEW_DB_URL
change DATABASE_URL
heroku pg:promote NEW_DB_URL
enable the app and workers
heroku maintenance:off
heroku ps:scale worker=1