How to show full post on home page

You can alter query_posts – see http://codex.wordpress.org/The_Loop – but maybe the best thing to do is use a page template with a new query. WP will default to home.php instead of index.php if one exists, so you can leave index.php alone for other page to use. See http://codex.wordpress.org/Class_Reference/WP_Query

Sample query; remove the permalink and/or title, if needed:

<?php $my_query = new WP_Query('cat=1&showposts=1'); ?>

<?php while ($my_query->have_posts()) : $my_query->the_post(); ?>

<a href="https://wordpress.stackexchange.com/questions/22117/<?php the_permalink() ?>" title="<?php the_title(); ?>">

<?php the_title(); ?></a>

<?php the_content(); ?>

<?php endwhile; ?>