Jenkins can do anything you can do given that the tools are installed and accessible. So an easy solution is to install docker and kubectl on Jenkins and provide him with the correct kube config so he can access the cluster. So if your host can use kubectl you can have a look at the $HOME/.kube/config file.

So in your job you can just use kubectl like you do from your host.

Regarding the images from Docker Hub:

Docker Hub is the default Docker Registry for Docker anyway so normally there is no need to change anything in your cluster only if you want to use your own Private Hosted Registry. If you are running your cluster at any cloud provider I would use there Docker registries because they are better integrated.

So this part of a deployment will pull nginx from Docker Hub no need to specify anything special for it:

    spec:
      containers:
      - name: nginx
      Image: nginx:1.7.9

So ensure Jenkins can do the following things from command line:

1. build Docker images
2. Push Docker Images (make sure you called docker login on Jenkins)
3. Access your cluster via kubectl get pods

So an easy pipeline needs to simply do this steps:

1. trigger on SVN change
2. checkout code
3. create a unique version which could be Build number, SVN Revision, Date)
4. Build / Test
5. Build Docker Image
6. tag Docker Image with unique version
7. push Docker Image
8. change image line in Kubernetes deployment.yaml to newly build version (if your are using Jenkins Pipeline you can use readYaml, writeYaml to achive this)
9. call kubectl apply -f deployment.yaml

Depending on your build system and languages used there are some useful tools which can help building and pushing the Docker Image and ensuring a unique tag. For example for Java and Maven you can use Maven CI Friendly Versions with any maven docker plugin or jib.