I am thinking about modifying a template inside a wordpress install to exhibit related posts ... during my situation, I believe I'd be fine with only showing the most recent 5 posts in the present post's category.

Wouldso would one do this without setting up a wordpress plugin? I'd rather customize the template in cases like this

I have poked around on the internet looking for and weren't effective at choosing the best API calls to achieve this.

There's an associated posts by category wordpress plugin, at http://wordpress.org/extend/plugins/related-posts-by-category/. Do the installation in Wordpress admin and modify your theme. The next code creates a number of five li's that contains links towards the posts.

<?php
related_posts_by_category(
      array(
        'orderby' => 'post_date',
        'order' => 'desc',
        'limit' => 5,
        'echo' => true,
        'before' => '<li>',
        'inside' => '',
        'outside' => '',
        'after' => '</li>',
    //  'rel' => 'nofollow',
        'type' => 'post',
    //    'message' => 'no matches',
        'showthis' => true
      )
    );
?>

Like pointed out through the other posters you'll need an wordpress plugin to make this happen. The main reason ist, that there's no API in wordpress to obtain the related posts. Exactly what the plug ins bsically do, is creating a database in the words in most posts and display the posts most abundant in common words.

See this question for many explanation how stackoverflow does this. It Programming Collective Intelligence includes a good explanation on a few of the more complex methods to the attached posts problem.