How to add a background image to category and display image on category page

You can try this code. It will add a media button which will get the image url. I have commented the parts that I have deleted or added.

function my_category_module()
{
    // this action is deprecated
    //add_action('edit_category_form_fields', 'add_image_cat');

    // Add these actions for edit and add
    add_action('edited_category', 'save_image');
    add_action('create_category', 'save_image');
    add_action('category_edit_form_fields', 'edit_image_cat');
}
add_action('init', 'my_category_module');


function edit_image_cat($tag)
{
    $category_images = get_option('category_images');
    $category_image="";
    if (is_array($category_images) && array_key_exists($tag->term_id, $category_images)) {
        $category_image = $category_images[$tag->term_id];
    }
    ?>
    <tr>
        <th scope="row" valign="top"><label for="auteur_revue_image">Image</label></th>
        <td>
            <?php
                if ($category_image != "") {
                    ?>
                <img src="https://wordpress.stackexchange.com/questions/352455/<?php echo $category_image; ?>" alt="" title="" />
            <?php
                }
                ?>
            <br />
            <!-- Add this html here -->
            <input type="text" class="regular-text" id="custom_category_image" name="category_image" value="https://wordpress.stackexchange.com/questions/352455/<?php echo $category_image; ?>">
            <button class="set_category_image button">Set Image url</button>

            <!--
                <input type="text" name="category_image" id="category_image" value="https://wordpress.stackexchange.com/questions/352455/<?php echo $category_image; ?>"/><br />
                <span>This field allows you to add a picture to illustrate the category. Upload the image from the media tab WordPress and paste its URL here.</span>
            -->
        </td>
    </tr>

<?php
}

function save_image($term_id)
{
    if (isset($_POST['category_image'])) {
        //load existing category featured option
        $category_images = get_option('category_images');
        //set featured post ID to proper category ID in options array
        $category_images[$term_id] =  $_POST['category_image'];
        //save the option array
        update_option('category_images', $category_images);
    }
}


// Enquey media elements
add_action('admin_enqueue_scripts', function () {
    if (is_admin())
        wp_enqueue_media();
});

// Add JS using admin_footer or enque thorugh hooks
add_action('admin_footer', 'my_footer_scripts');
function my_footer_scripts()
{
    ?>
    <script>
        jQuery(document).ready(function() {
            if (jQuery('.set_category_image').length > 0) {
                if (typeof wp !== 'undefined' && wp.media && wp.media.editor) {
                    jQuery('.set_category_image').on('click', function(e) {
                        e.preventDefault();
                        var button = jQuery(this);
                        var url_input = jQuery("#custom_category_image");
                        wp.media.editor.send.attachment = function(props, attachment) {
                            url_input.val(attachment.url);
                        };
                        wp.media.editor.open(button);
                        return false;
                    });
                }
            }
        });
    </script>
<?php
}