Take a look at handler lifecycle events, depending on the kind of health check probe you are doing the handler event will vary. I tried this on a simple "hello-world" pod. Set up a command in your postStart event, and this will run only once, when the container is built and running. My example yaml file looks like this:

apiVersion: apps/v1
kind: Deployment
metadata:
  name: hello-world
  labels:
    app: hello-world
spec:
  replicas: 1
  selector:
    matchLabels:
      app: hello-world
  template:
    metadata:
      labels:
        app: hello-world
    spec:
      containers:
      - image: paulbouwer/hello-kubernetes:1.5
        lifecycle:
          postStart:
            exec:
              command: ["/bin/sh", "-c", "echo Hello from the postStart handler > /usr/share/message"]
        name: hello-world
        ports:
        - containerPort: 8080
          name: http