Show featured image as background image from postQuery

You are correct, you will need to get the image URL and set is as a variable so you can echo it out as an inline style in your HTML. I’ve updated your foreach loop below and set the image as an inline style of your <li> element:

foreach( $postQuery as $post ) : setup_postdata($post);
    if ( has_post_thumbnail() ) { ?>
        /* Get the post thumbnail source so we can get the exact URL of the image.  This returns an array of stuff for us */
        $thumbImg = wp_get_attachment_image_src( get_post_thumbnail_id($post->ID), 'feature-slider' );
        /* Since we want the URL, let's get the first value from our new array and set it as $thumbImgUrl */
        $thumbImgUrl = $thumbImg['0']; ?>
        <li class="frontSlider" style="background-image: url('<?php echo $thumbImgUrl; ?>');">
            <a href="https://wordpress.stackexchange.com/questions/225085/<?php echo get_permalink(); ?>" title="Go to <?php echo the_title(); ?>" rel="bookmark">
                <div class="flex-captionWrap">
                    <p class="flex-caption"><?php echo get_post(get_post_thumbnail_id())->post_excerpt; ?></p><!-- Retrieves text string from Captions field in Media -->
                </div>
            </a>
        </li>
    <?php 
    }
endforeach; ?>

Hope that helps! I did not test this before posting it, but I’ve done this several times before so it should work just fine.