Show category name, and last 5 post titles below category name?

you haven’t stated where the category id will come from, nor how you want to style the category title, and if the post titles are supposed to be linked to the single post; therefore just the basic structure:

<?php 
$cat_id = 137; 
    echo '<div class="category-title">'.get_category($cat_id)->name.'</div>;
    $cat_posts = get_posts(array('category__in' => array($cat_id), 'posts_per_page' =>5));
    if($cat_posts) {
        echo '<ul class="category-posts">';
        foreach($cat_posts as $cat_post) {
            echo '<li class="category-post">'.get_the_title($cat_post->ID).'</li>';
        }
        echo '</ul>';
    }
?>