Setting up CI/CD for .NET Core in Docker?

Question

I'm new to CI/CD on Gitlab and would like some help filling in the gaps for setting up a pipeline for a Visual Studio solution containing multiple .NET Core projects.

I'm hoping for a process something like this:

1. Pass in a PROJECT_NAME variable to the CI runner which specifies which project in the solution to build/publish.
2. The runner builds the image using the .NET Core 3.1 SDK image.
3. The runner publishes the image to the container registry.
4. The runner fires off a webhook to a predefined webhook URL on the target server where the image should be deployed.
5. The webhook triggers a redeploy of the container on the host with the new image.

This is what I've put together so far for my .gitlab-ci.yml file but I'm not sure if this is correct or what to do next?

stages:
    - build
    - publish
  
  build:
    stage: build
    image: mcr.microsoft.com/dotnet/core/sdk:3.1-buster
    only:
      - master
    script:
      - cd $PROJECT_NAME
      - dotnet restore
      - dotnet build -c Release
      - dotnet publish -c Release
    artifacts:
      paths:
        - ./bin/Release/netcoreapp3.1/publish/*
      expire_in: 1 week
    tags:
    - docker
  
  publish:
    stage: publish
    image: docker:stable
    services:
      - docker:dind
    only:
      - master
    before_script:
      - docker login registry.gitlab.com -u $CI_REGISTRY_USER -p $CI_REGISTRY_PASSWORD
    script:
      - docker build -t $CI_REGISTRY/$PROJECT_NAME .
      - docker push $CI_REGISTRY/$PROJECT_NAME
    after_script:
      - docker logout $CI_REGISTRY
      - curl -H "Authorization: Bearer $WEBHOOK_ACCESS_TOKEN" $WEBHOOK_URL
    tags:
    - docker

Any pointers would be appreciated.

Thanks in advance.