Is there a simple way to deploy a local build kubernetes on AWS?

If you already have an existing cluster running, then the answer is to either push your images to a common docker registry (either hub.docker.com, or ECR, or a locally hosted registry, or whatever), or to just cheat and for h in $(cluster node addresses); do docker save my-kubernetes-image:my-tag | ssh $h docker load; done, then after you have made the images available, update all the manifests to point to your new image.

If you don't already have a cluster, then I would suspect any of the existing toys, even the repugnant kops, would get you a cluster, then you can swap out the images and do so as many times as is required to verify your fix.