How to bring a docker data container to Kubernetes?
You'll want to look through the Kubernetes Volume documentation to see what options are available.

In particular look into the PersistentVolumeClaim.  This allows you to setup a shared data volume for your pods.  Here's an example snippet from a k8s deployment YAML using a volume:

spec:
      containers:
      - image: **yourdbcontainer**
        name: mysql-db
        env:
        - name: example
          value: example_value
        ports:
        - containerPort: 3306
          name: mysql-db
        volumeMounts:
        - name: mysql-persistent-storage
          mountPath: /var/lib/mysql
      volumes:
      - name: mysql-persistent-storage
        persistentVolumeClaim:
          claimName: mysql-pv-claim