On your Synology NAS, create a "docker" folder on your shared volume and then create a subfolder "homebridge". After creating your config.json and package.json files, copy them to the shared "docker/homebridge" folder. The config.json will have your homebridge config and the package.json will list the NPM packages that you want to be built each time the container is run. See https://github.com/cbrandlehner/homebridge-docker/tree/master/config-sample for samples of both the config.json and package.json files.
Run the container from the homebridge_start.sh via the Synology Task Scheduler or via SSH command line by entering
sudo docker run --name=homebridge -d --restart=always --net=host -p 51826:51826 -p 8080:8080 -v /volume1/docker/homebridge:/root/.homebridge cbrandlehner/homebridge:0.19
Enter your root password to run the command.
If you want to use Synology Task Scheduler in the Control Panel of the NAS, you will need to copy homebridge_start.sh to the "docker/homebridge" folder on the shared volume. Next, create a task that runs the docker/homebridge/homebridge_start.sh at bootup and optionally, I have another that restarts the container at 1:45am everynight. This is not neccesary, but in my earlier testing, I often rebooted and just built the task to keep it all clean. For the tasks, you will need to make the task user root.
If you are running the firewall on the Synology, you will need to open TCP 5353 & 51826 & 8080. You could select from a list of built-in applications and select "Docker homebridge 51826 Docker (TCP)" and "Bonjour 5353 Bonjour Service" You can find the firwall profiles under the Control Panel/Security. If you do not open these ports, the firewall will close within a few minutes and your app will not be reachable from an IOS device.