Displaying a custom field in Category Archives

If your Loop is constructed correctly $post will be set for each post in the loop in turn. You could also consider using get_the_ID.

get_post_meta($post->ID, "user_submit_url", true); by itself won’t do much. You will have to format and echo the information but that should retrieve your post meta data.

For example, to generate a link (assuming your meta value is an URL and that you’ve save a string and not an array):

$meta = get_post_meta($post->ID,'user_submit_url',true); 
echo '<a class="meta" href="'.$meta.'" target="_blank">'.$meta.'</a>';

If your theme has a category.php and you want this only on category archives put that code in category.php. archive.php is used by all archives that do not have a specialized template– tag and author archives, for example. The template hierarchy is explained in the Codex.