Good question! So first I build the docker container with a Dockerfile I keep at the root of my project. I also have an aws private docker repo. (Read more about it here: https://aws.amazon.com/ecr/). So in my gitlab-ci I have a statement like the following:

  - docker build -t container-name:$CI_BUILD_REF_NAME .
  - docker tag container-name:$CI_BUILD_REF_NAME $AWS_ACCOUNT_ID.$AWS_ACCOUNT_REGION.amazonaws.com/container-name:$CI_BUILD_REF_NAME
  - docker push $AWS_ACCOUNT_ID.$AWS_ACCOUNT_REGION.amazonaws.com/container-name:$CI_BUILD_REF_NAME

This builds my container, tags it, and pushes it to my ec2 container registry. I do most of my deployments using AWS ECS (EC2 Container Service, you can read more about it here: https://aws.amazon.com/ecs/)

My secret sauce is in the deployment stage of my ci. I basically run a command which deploys to a ECS environment. This is a custom script which is part of my overall AWS process. I have scripted out the creation of everything on AWS (from building the environment, spinning up EC2's, security and security groups, auto scaling groups, container registry, deploying jobs into an ECS). In the current project I am in we are using Ruby so the command is just a rake task and looks like:

  stages:
   - test
   - build
   - deploy
  
  docker_deploy:
    stage: deploy
    script:
     - do some stuff
     - rake awsenv:redeploy[$CI_BUILD_REF_NAME]
    tags:
     - general
    only:
     - development
     - staging
     - master

The rake task logs into the awsapi securely, runs some commands, verifies the work, and calls the task complete in gitlab-ci. If you have any further questions, feel free to email me. It's my username at gmail.