In functions.php in your child theme, you can do a couple different things on the login screen. First, you’ll want to add a custom stylesheet by hooking into login_enqueue_scripts.
add_action( 'login_enqueue_scripts', 'wpse_login_styles' );
function wpse_login_styles() {
wp_enqueue_style( 'wpse-custom-login', get_stylesheet_directory_uri() . '/style-login.css' );
}
Then you can create a styles-login.css file in the root of your child theme directory (or adjust the path in wp_enqueue_script based on your needs).
Once you’ve made it that far, you can customize the login page however you’d like using CSS in that file. For example, change the login logo with these instructions ripped from the Codex:
#login h1 a, .login h1 a {
background-image: url(path/to/login/logo.png);
padding-bottom: 30px;
}
Beyond CSS, you can also change the URL and site title on this page using the login_headerurl and login_headertitle hooks, also described in the above Codex link.
If you want to go further like adding elements to the page, or even creating a custom login page, that’s all described here and here.