One way of doing this would be to create a "CSS" file from PHP. In other words, create a file, call it something like
style.css.php, and at the beginning of the file put:
<?php header("Content-type: text/css");
Then, link that file in the head of your theme file. Because the
style.css.php file is a PHP file, anything you can do in a normal PHP file can be done in this file. As such, you can pull theme option values from the database and use them. For example:
background: <?php echo get_option('my-header-background-color'); ?>
Of course, you need to get the options into the database first, but since your question didn't address that, I won't go into it here.