You should add the libpng-dev package to your Dockerfile:

FROM php:5.6-apache

RUN docker-php-ext-install mysql mysqli

RUN apt-get update -y && apt-get install -y sendmail libpng-dev

RUN apt-get update && \
    apt-get install -y \
        zlib1g-dev 

RUN docker-php-ext-install mbstring

RUN docker-php-ext-install zip

RUN docker-php-ext-install gd

Then go to directory with Dockerfile and run:

docker build -t sitename .

It worked in my case:

Removing intermediate container f03522715567
Successfully built 9d69212196a2

Let me know if you get any errors.

EDIT:

You should see something like this:

REPOSITORY          TAG                 IMAGE ID            CREATED             SIZE
sitename            latest              9d69212196a2        19 minutes ago      414 MB
<none>              <none>              b6c69576a359        25 minutes ago      412.3 MB

EDIT2:

Just to double check everything:

Please run the docker build command this way:

docker build -t sitename:1.0 .

(adding :1.0 should not change anything, I added it just to have additional row in docker images output)

Then start the container:

docker run --name sitename_test -p 80:80 sitename:1.0

It should work just fine.

I assumed that apache is using standard port (80) - maybe you need to adjust that. If you have other services/containers listening on port 80 you can make your container listening on other port:

docker run --name sitename_test -p 8080:80 sitename:1.0

That will redirect the traffic from port 8080 to port 80 "inside" the container.

Normally you run container in the background. To do this add the -d option to the docker run command (but for testing purposes you can omit -d to see output in the console).

If you decide to run container in the background you can check logs using docker logs sitename_test. To follow the logs (and see updates in logs) use -f option:

docker logs -f sitename_test

Here's how I do it but on php image, not alpine. Maybe you can just "finetune" it to work on alpine:

RUN apt-get update && \
    apt-get install -y libfreetype6-dev libjpeg62-turbo-dev libpng12-dev && \
    docker-php-ext-configure gd --with-freetype-dir=/usr/include/ --with-jpeg-dir=/usr/include/ && \
    docker-php-ext-install gd

You can add these settings in your Dockerfile:

FROM php:7.4-fpm
RUN apt-get update && apt-get install -y \
        libfreetype6-dev \
        libjpeg62-turbo-dev \
        libpng-dev \
    && docker-php-ext-configure gd --with-freetype --with-jpeg \
    && docker-php-ext-install -j$(nproc) gd

Official documentation.

following this answer install gd from dockerfile

RUN docker-php-ext-install gd 

https://github.com/rhamdeew/docker-php-8-fpm-alpine/blob/master/Dockerfile