How to hook into the subscriber /wp-admin/index.php page?

All of WP’s built-in roles have subscriber capabilities. You’re basically checking here, “if the current user has at least subscriber capabilities”. If you want to isolate only the subscribers, you can check for the absence of a higher capability. If you are just using built-in roles, the next higher role of Contributor has edit_posts and delete_posts capabilities by default.

if(!current_user_can('edit_posts')) {
    // display your message
}

As far as adding content this way, admin_init is a pretty early hook, so depending on your content and how it is styled, it may cause some layout quirks. You mentioned you don’t want to use a dashboard widget, so you might want to look into other hooks, or perhaps add an options page that subscribers can view with your custom content.