How to connect to local MySQL server through Docker?

The Docker CLI docs give this solution (which assumes you are running on a Linux host with ):

Sometimes you need to connect to the Docker host from within your container. To enable this, pass the Docker host’s IP address to the container using the --add-host flag. To find the host’s address, use the ip addr show command.

The flags you pass to ip addr show depend on whether you are using IPv4 or IPv6 networking in your containers. Use the following flags for IPv4 address retrieval for a network device named eth0:

$ HOSTIP=`ip -4 addr show scope global dev eth0 | grep inet | awk '{print \$2}' | cut -d / -f 1`
$ docker run  --add-host=docker:${HOSTIP} --rm -it debian

Then the name docker inside the container will map to the host's IP address. For your case, you could use docker run --add-host=mysql_server:$(hostip) ...

If using Boot2Docker, it sets up a mapping to the host at a predefined address, so on that platform the equivalent to the above is just the one command:

$ docker run  --add-host=docker:192.168.59.3 --rm -it debian

To connect local MySQL you can definitely use --network="host" in your docker run command.
Then 127.0.0.1 or localhost in your docker container will point to your docker host.

docker run --network="host" -p 8080:8080 <your-docker-Image>