how to show posts on different pages if number of post per page is one

First of all, never use query_posts(), use WP_Query() instead. (Learn More)

With your WP_Query() use paged parameter in the argument array, like:

<?php
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$args = array(
  'posts_per_page' => 1,
  'paged' => $paged
);

$cat_query = new WP_Query( $args );
?>

And for the pagination link, go with:

<?php
$total_pages = $wp_query->max_num_pages;
if ( $total_pages > 1 ) { ?>

    <div id="nav-below" class="navigation">

        <div class="nav-previous">
            <?php next_posts_link(__( '<span class="meta-nav">&laquo;</span> Older Entries', 'your-theme' )) ?>
        </div>
        <div class="nav-next">
            <?php previous_posts_link(__( 'Newer Entries <span class="meta-nav">&raquo;</span>', 'your-theme' )) ?>
         </div>

    </div><!-- #nav-below -->

<?php } //endif ( $total_pages > 1 )

And don’t forget to use wp_reset_postdata() to reset the query after the necessary code block.