How do you turn a custom field’s value into a permalink for that post?

Use make_clickable() function:

add_action( 'genesis_entry_footer', 'custom_comments_field' );
function custom_comments_field() {
    if (  is_single() && genesis_get_custom_field('Comments link text') ) :
        echo '<div class="comments-link">'. make_clickable( genesis_get_custom_field('Comments link text') ) .'</div>';
    endif;
}

UPDATE:

In that case, try this:

add_action( 'genesis_entry_footer', 'custom_comments_field' );
function custom_comments_field() {
    if (  is_single() && genesis_get_custom_field('Comments link text') ) :
        global $post;
        echo '<div class="comments-link"><a href="'. get_permalink( $post->ID ) .'">'. genesis_get_custom_field('Comments link text') .'</a></div>';
    endif;
}