If you add an image to your category description, and set the module parameter "Category Description" to yes, the description (if any) AND the image will be displayed in your module without any overrides.
If you need to add the category image somewhere else (e.g. above the title), you could try this in your override file template/my-template/html/mod_articles_categories/
:
<?php
$string = $item->description;
$beg = substr($string,strpos($string,"<img"),strlen($string));
//$catimg = substr($beg,0,$pos_srch = strpos($beg,">"));
$catimg = substr($beg,0,$pos_srch = strpos($beg,">"));
echo $catimg;
?>
This will echo the first image from the category description, including the <img>
tag.