Docker In Kubernetes Deployment
Unverified as it sounds brittle to me to start a container outside of k8s supervision, but you should be able to mount /var/run/docker.sock with a hostPath volume.

Example variation from the documentation:

apiVersion: v1
kind: Pod
metadata:
  name: test-pd
spec:
  containers:
  - image: gcr.io/google_containers/test-webserver
    name: test-container
    volumeMounts:
    - mountPath: /var/run/docker.sock
      name: docker-sock-volume
  volumes:
  - name: docker-sock-volume
    hostPath:
      # location on host
      path: /var/run/docker.sock
      # this field is optional
      type: File


I think a simple mount should be enough to allow communication from docker client within the container to docker daemon on host but in case you get a write permission error it means you need to run your container as privileged container
 using a securityContext object like such (just an extract from above to show the addition, values taken from the documentation):

spec:
  containers:
  - image: gcr.io/google_containers/test-webserver
    securityContext:
      privileged: true
    name: test-container