How to automate TeamCity installation using docker? JetBrains TeamCity docker image describe build steps for already installed TeamCity.

Is there any way to automate installation of TeamCity it-self?

For example creating the admin account, enabling standard authentication and setting up the project?
Automation of installations like this can often be done with config management tools.  GitHub and the public repos for these tools is a good place to start.

There is a Puppet installer for TeamCity here: https://github.com/haf/puppet-teamcity

Here's a Chef cookbook:
https://supermarket.chef.io/cookbooks/chef-teamcity

You can also use tools like Packer to generate your image(s).  Packer can substitute for a Dockerfile in generating a Docker image. Packer will also run Provisioners (including Chef/Puppet/Ansible) to configure your image.