How to connect IntelliJ with local MySQL?

Connecting to a local instance is essentially the same as connecting to a remote instance of MySQL. Just substitute either localhost, or 127.0.0.1 in place of the IP address you would use normally.

To add a new database connection (called a data source in IntelliJ), open the Database window View -> Tool Windows -> Databases, then click the + sign and select Data Source and then MySQL from the sub-menu. The defaults for the MySQL connection should for a local install of MySQL.

To open a connection, right click on your new data source and select Open Console.

Official IntelliJ Documentation - Managing Data Sources

As of Community version 2017.2, the DB Browser does not come bundled with the IDE (at least not on my last two installations). In order to activate it, you should navigate in the IDE in **File->Settings->Plugins->Browse repositories and select "Database" from the dropdown menu. From there you can install the Database Navigator. After the installation has been successful, you should restart the IDE. Then you can select **View->Tool windows->DB Browser.