How do I use just one category’ posts on my home page?

This should about do it for you, add args as needed

$args = array(
    //basic post data
    'posts_per_page' => 10,
    'post_type'      => 'post',
    'post_status'    => 'publish',
    //category
    'category_name'  => 'your-cat-slug'
);
$posts = new WP_Query( $args );

Docs: WP_Query