While trying to install Bloodhound on my new Kali (Well I have to reinstall my Kali several times due to the conflicted tools). I found no actual guidance is available online on the Internet. So let me help you out:

# Update the packages and upgrade your system first
sudo apt update && sudo apt upgrade -y
# Install Docker
sudo apt install docker.io
# Don't forget to add yourself to docker group
sudo usermod -aG docker $USER
# You should restart at this point

At this point Docker is usable, but your Kali will keep sending error messages due to the non-existence of Kali docker package, therefore you need to switch it to Debian.

# Use whatever editor you like
sudo nvim /etc/apt/sources.list.d/docker.list
 
# The initial configuration:
deb [arch=arm64 signed-by=/etc/apt/keyrings/docker.asc] https://download.docker.com/linux/debian kali-rolling stable
 
# The modified configuration:
deb [arch=arm64 signed-by=/etc/apt/keyrings/docker.asc] https://download.docker.com/linux/debian bookworm stable

I am not sure whether it will be any different in the future, but this does work right now.

# Install docker-compose
sudo apt update
sudo apt install docker-compose-plugin

Enjoy your Docker.