How do I hook my function to run only after submitting login form

The filter runs on every page load, but the first argument will be a WP_User object only on successful submission. It will be null if nothing was submitted, but most likely it will be a WP_Error object, which you can check for. If you want to know the specific error you have to look at the errors the object contains. The case where the form was not submitted will have both empty_username and empty_password error array keys.

This will possibly shed some light on things:

function myplugin_auth_signon( $user, $username, $password ) {
    var_dump( $user );
    return $user;
}
add_filter( 'authenticate', 'myplugin_auth_signon', 30, 3 );

Load the page, then try various combinations of incorrect and empty values, then a correct login.