Your problem is not the connection between your containers. The problem is your PHP / Apache container which doesn't support mysqli (or PDO MySQL). WordPress can't find another function to connect with a MySQL database or your MySQL container. Instead WordPress is using a deprecated and removed (since PHP 7.0) mysql_ function per default. You need to install at least mysqli on your PHP container (explained below).
I also recommend to use a docker-compose file to install and run all containers with one command.
To create the containers you want, you can use the following docker-compose.yml file:
version: "3"
services:
  achi-php-apache:
    build:
      context: ./
    container_name: achi-php-apache
    ports:
      - "8080:80"
    volumes:
      - /home/achi/workspace/web/wordpress-template:/var/www/html:rw
    depends_on:
      - achi-mysql
    networks:
      - wp-net
  achi-mysql:
    image: mysql:5.7
    container_name: achi-mysql
    ports:
      - "3306:3306"
    environment:
      MYSQL_ROOT_PASSWORD: 123456
      MYSQL_DATABASE: wp-dbname
    volumes:
      - wp-mysql-data:/var/lib/mysql
    networks:
      - wp-net
networks:
  wp-net:
    driver: bridge
volumes:
  wp-mysql-data:
You need the following Dockerfile on the same directory as the docker-compose.yml file:
FROM php:7.0-apache
RUN docker-php-ext-install -j$(nproc) mysqli
This Dockerfile installs the missing mysqli extension so WordPress can use it.
You can also use PDO MySQL instead of mysqli. In this case you can use the following Dockerfile:
FROM php:7.0-apache
RUN docker-php-ext-install -j$(nproc) pdo
RUN docker-php-ext-install -j$(nproc) pdo_mysql
Now you can execute the command docker-compose up inside the folder where the docker-compose.yml file is located. After creating the container and running you should be able to access the WordPress site (<ip-or-hostname>:8080).
On the wp-config.php file you need to use the following constants:
define('DB_NAME', 'wp-dbname');
define('DB_USER', 'root');
define('DB_PASSWORD', '123456');
define('DB_HOST', 'achi-mysql');
You can also use the official WordPress image to install WordPress. In this case you can use the following docker-compose.yml file:
version: "3"
services:
  achi-php-apache:
    image: wordpress:4.9.4-php7.0-apache
    container_name: achi-php-apache
    ports:
      - "8080:80"
    environment:
      WORDPRESS_DB_HOST: achi-mysql
      WORDPRESS_DB_USER: root
      WORDPRESS_DB_PASSWORD: 123456
      WORDPRESS_DB_NAME: wp-dbname
    volumes:
      - /home/achi/workspace/web/wordpress-template:/var/www/html:rw
    depends_on:
      - achi-mysql
    networks:
      - wp-net
  achi-mysql:
    image: mysql:5.7
    container_name: achi-mysql
    ports:
      - "3306:3306"
    environment:
      MYSQL_ROOT_PASSWORD: 123456
      MYSQL_DATABASE: wp-dbname
    volumes:
      - wp-mysql-data:/var/lib/mysql
    networks:
      - wp-net
networks:
  wp-net:
    driver: bridge
volumes:
  wp-mysql-data: