Custom Sign Up Form for a Custom WordPress Website

Maybe this plugins can help you
https://wordpress.org/plugins/pagerestrict/
https://wordpress.org/plugins/wordpress-access-control/

Update

if ( is_user_logged_in() ) {
    // you content
}
else {
    // you text
}

UPDATE 2

When the user is authorized, him ID was write in global array. This ID you can take from this array.

$current_user = wp_get_current_user();

I don’t know structure your table, but you need to have user ID field like in wp_users table. Make a request to the database.

global $wpdb

$value = $wpdb->get_result("SELECT `user_name_in_your_table`
                            FROM `wp_registration`
                            WHERE `user_id` = '$current_user->ID'");

if($value[0]->user_name_in_your_table !== ''){
    //access
} else{
    //denied
}