EDIT - since Jul 23, 2015

The official postgres docker image will run .sql scripts found in the /docker-entrypoint-initdb.d/ folder.

So all you need is to create the following sql script:

init.sql

CREATE USER docker;
CREATE DATABASE docker;
GRANT ALL PRIVILEGES ON DATABASE docker TO docker;

and add it in your Dockerfile:

Dockerfile

FROM library/postgres
COPY init.sql /docker-entrypoint-initdb.d/

But since July 8th, 2015, if all you need is to create a user and database, it is easier to just make use to the POSTGRES_USER, POSTGRES_PASSWORD and POSTGRES_DB environment variables:

docker run -e POSTGRES_USER=docker -e POSTGRES_PASSWORD=docker -e POSTGRES_DB=docker library/postgres

or with a Dockerfile:

FROM library/postgres
ENV POSTGRES_USER docker
ENV POSTGRES_PASSWORD docker
ENV POSTGRES_DB docker

for images older than Jul 23, 2015

From the documentation of the postgres Docker image, it is said that

[...] it will source any *.sh script found in that directory [/docker-entrypoint-initdb.d] to do further initialization before starting the service

What's important here is "before starting the service". This means your script make_db.sh will be executed before the postgres service would be started, hence the error message "could not connect to database postgres".

After that there is another useful piece of information:

If you need to execute SQL commands as part of your initialization, the use of Postgres single user mode is highly recommended.

Agreed this can be a bit mysterious at the first look. What it says is that your initialization script should start the postgres service in single mode before doing its actions. So you could change your make_db.ksh script as follows and it should get you closer to what you want:

NOTE, this has changed recently in the following commit. This will work with the latest change:

export PGUSER=postgres
psql <<- EOSQL
    CREATE USER docker;
    CREATE DATABASE docker;
    GRANT ALL PRIVILEGES ON DATABASE docker TO docker;
EOSQL

Previously, the use of --single mode was required:

gosu postgres postgres --single <<- EOSQL
    CREATE USER docker;
    CREATE DATABASE docker;
    GRANT ALL PRIVILEGES ON DATABASE docker TO docker;
EOSQL

Have you considered using the POSTGRES_DB environment variable ?

services:
  db:
    container_name: postgres
    image: postgres
    environment:
      POSTGRES_USER: ${POSTGRES_USER}
      POSTGRES_PASSWORD: ${POSTGRES_PASSWORD}
      PGDATA: /data/postgres
      POSTGRES_DB: bank
    volumes:
      - db:/data/postgres
    ports:
      - "5332:5432"
    networks:
      - db
    restart: unless-stopped
    healthcheck:
      test: [ "CMD-SHELL", "pg_isready -d postgres" ]
      interval: 30s
      timeout: 10s
      retries: 5
networks:
  db:
    driver: bridge

volumes:
  db: