How can I browse my Heroku database?

Try heroku pg:psql. For instructions for how to do this, see this.

If you want to browse your Heroku PostgreSQL database using a browser there is now a free tool for that called Heroku Data Explorer: https://datazenit.com/heroku-data-explorer.html

It uses Heroku API in real time so your data/credentials stay secure and private. Also it has excellent performance because Heroku local network is used for database connections.