Next Step: Alpine is multi-arch

As of today, the multi-arch manifest of alpine points to several architectures, including s390x. This means, you can now re-use all the Dockerfiles saying "FROM alpine" without changes (no "s390x/"). Or, of course, things like "docker run -ti alpine sh" work without an s390x/.

Portainer

Portainer is one of the major Open Source tools for graphically managing Docker environments. You can use portainer on any machine and direct it to a Docker API endpoint to manage s390x.

Update 2017/11/08: starting today, portainer/portainer comes with s390x support. Check out containerz.blogspot.com/2017/11/portainer-revisited.html for details; the steps below are not required anymore to run portainer.

If you prefer running it on s390x, there are some steps needed, as long as portainer doesn't build it for s390x:

Docker and Kubernetes. Kubernetes and Docker.

Kubernetes has become the trending orchestration solution for containers. All big Cloud providers bet on it, and even in smaller companies, k8s (written that way as there are 8 letters between k and s) is what all the hip developers go for.
Docker is still used to build all the images, and they have made it to a commercial product used by many customers.
Kubernetes has focused on being an extensible and scalable framework, and is still growing fast; it has a credible reputation for managing data center scale with best possible control. In fact, control is IMHO the word that describes its nature best.
Docker has chosen to put user experience first: it provides rich functionality with sane defaults, but users don't have to think about it -- Docker simply and quickly does what you mean. User experience is (IMHO) Docker's core characteristics.