Add Date & Author Meta to Category Archive Page

This is how its done in Twenty Fourteen which uses a template tag:

Note: The conditionals if ( is_sticky() && is_home() && ! is_paged() ) { will need to be changed or removed

if ( ! function_exists( 'twentyfourteen_posted_on' ) ) :
/**
 * Print HTML with meta information for the current post-date/time and author.
 *
 * @since Twenty Fourteen 1.0
 */
function twentyfourteen_posted_on() {
    if ( is_sticky() && is_home() && ! is_paged() ) {
        echo '<span class="featured-post">' . __( 'Sticky', 'twentyfourteen' ) . '</span>';
    }

    // Set up and print post meta information.
    printf( '<span class="entry-date"><a href="https://wordpress.stackexchange.com/questions/166478/%1$s" rel="bookmark"><time class="entry-date" datetime="%2$s">%3$s</time></a></span> <span class="byline"><span class="author vcard"><a class="url fn n" href="%4$s" rel="author">%5$s</a></span></span>',
        esc_url( get_permalink() ),
        esc_attr( get_the_date( 'c' ) ),
        esc_html( get_the_date() ),
        esc_url( get_author_posts_url( get_the_author_meta( 'ID' ) ) ),
        get_the_author()
    );
}
endif;

The template tag twentyfourteen_posted_on(); is added to content.php file below the entry-title

<?php
/**
 * The default template for displaying content
 *
 * Used for both single and index/archive/search.
 *
 * @package WordPress
 * @subpackage Twenty_Fourteen
 * @since Twenty Fourteen 1.0
 */
?>

<article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
    <?php twentyfourteen_post_thumbnail(); ?>

    <header class="entry-header">
        <?php if ( in_array( 'category', get_object_taxonomies( get_post_type() ) ) && twentyfourteen_categorized_blog() ) : ?>
        <div class="entry-meta">
            <span class="cat-links"><?php echo get_the_category_list( _x( ', ', 'Used between list items, there is a space after the comma.', 'twentyfourteen' ) ); ?></span>
        </div>
        <?php
            endif;

            if ( is_single() ) :
                the_title( '<h1 class="entry-title">', '</h1>' );
            else :
                the_title( '<h1 class="entry-title"><a href="' . esc_url( get_permalink() ) . '" rel="bookmark">', '</a></h1>' );
            endif;
        ?>

        <div class="entry-meta">
            <?php
                if ( 'post' == get_post_type() )
                    twentyfourteen_posted_on();

                if ( ! post_password_required() && ( comments_open() || get_comments_number() ) ) :
            ?>
            <span class="comments-link"><?php comments_popup_link( __( 'Leave a comment', 'twentyfourteen' ), __( '1 Comment', 'twentyfourteen' ), __( '% Comments', 'twentyfourteen' ) ); ?></span>
            <?php
                endif;

                edit_post_link( __( 'Edit', 'twentyfourteen' ), '<span class="edit-link">', '</span>' );
            ?>
        </div><!-- .entry-meta -->
    </header><!-- .entry-header -->

    <?php if ( is_search() ) : ?>
    <div class="entry-summary">
        <?php the_excerpt(); ?>
    </div><!-- .entry-summary -->
    <?php else : ?>
    <div class="entry-content">
        <?php
            the_content( __( 'Continue reading <span class="meta-nav">&rarr;</span>', 'twentyfourteen' ) );
            wp_link_pages( array(
                'before'      => '<div class="page-links"><span class="page-links-title">' . __( 'Pages:', 'twentyfourteen' ) . '</span>',
                'after'       => '</div>',
                'link_before' => '<span>',
                'link_after'  => '</span>',
            ) );
        ?>
    </div><!-- .entry-content -->
    <?php endif; ?>

    <?php the_tags( '<footer class="entry-meta"><span class="tag-links">', '', '</span></footer>' ); ?>
</article><!-- #post-## -->

Then the content.php file is added to the category.php file using get_template_part( 'content', get_post_format() );

<?php
/**
 * The template for displaying Category pages
 *
 * @link http://codex.wordpress.org/Template_Hierarchy
 *
 * @package WordPress
 * @subpackage Twenty_Fourteen
 * @since Twenty Fourteen 1.0
 */

get_header(); ?>

    <section id="primary" class="content-area">
        <div id="content" class="site-content" role="main">

            <?php if ( have_posts() ) : ?>

            <header class="archive-header">
                <h1 class="archive-title"><?php printf( __( 'Category Archives: %s', 'twentyfourteen' ), single_cat_title( '', false ) ); ?></h1>

                <?php
                    // Show an optional term description.
                    $term_description = term_description();
                    if ( ! empty( $term_description ) ) :
                        printf( '<div class="taxonomy-description">%s</div>', $term_description );
                    endif;
                ?>
            </header><!-- .archive-header -->

            <?php
                    // Start the Loop.
                    while ( have_posts() ) : the_post();

                    /*
                     * Include the post format-specific template for the content. If you want to
                     * use this in a child theme, then include a file called called content-___.php
                     * (where ___ is the post format) and that will be used instead.
                     */
                    get_template_part( 'content', get_post_format() );

                    endwhile;
                    // Previous/next page navigation.
                    twentyfourteen_paging_nav();

                else :
                    // If no content, include the "No posts found" template.
                    get_template_part( 'content', 'none' );

                endif;
            ?>
        </div><!-- #content -->
    </section><!-- #primary -->

<?php
get_sidebar( 'content' );
get_sidebar();
get_footer();

You could add the code directly to your category.php if you like.

    // Set up and print post meta information.
    printf( '<span class="entry-date"><a href="https://wordpress.stackexchange.com/questions/166478/%1$s" rel="bookmark"><time class="entry-date" datetime="%2$s">%3$s</time></a></span> <span class="byline"><span class="author vcard"><a class="url fn n" href="%4$s" rel="author">%5$s</a></span></span>',
        esc_url( get_permalink() ),
        esc_attr( get_the_date( 'c' ) ),
        esc_html( get_the_date() ),
        esc_url( get_author_posts_url( get_the_author_meta( 'ID' ) ) ),
        get_the_author()
    );