Display categories followed by their posts

How could I possibly display a list of category titles and their posts below them on the index.php.

Every section DIV should be a category, with a LI list of every post.

<?php get_header(); ?>
<!-- START CATEGORY -->
<div class="section">
<!-- CATEGORY TITLE -->
<h2><?php
$category = get_the_category();
echo $category[0]->cat_name;
?></h2>

<?php while ( have_posts() ) : the_post() ?>
<!-- START LIST OF POSTS IN THE CATEGORY -->
• <div class="header">
  <div class="checkbox">
  <input class="checkj" type="checkbox" id="c<?php the_ID(); ?>" name="cc" />
  <label for="c<?php the_ID(); ?>"><span></span></label>
  </div>
  <h4><?php the_title(); ?><?php
  $args = array( 'post_type' => 'attachment', 'posts_per_page' => -1, 'post_status' =>'any', 'post_parent' => $post->ID );
  $attachments = get_posts( $args );
  if ( $attachments ) {
  foreach ( $attachments as $attachment ) {
  the_attachment_link( $attachment->ID , false );
  echo '<span>' . $attachment->post_date . '</span>';
  }
  }
  ?></h4>
  </div>
  <?php the_content(); ?>

<?php endwhile; ?>

</div>
</div>
<?php get_footer(); ?>