How can I display only the post titles from a selected category in columns?

Updated answer:
Use two floated lists to emulate columns, same approach as previously though.

<?php
/*
Template Name: PageOfPosts
*/

get_header(); ?>

<div id="content">
    <div class="t">
        <div class="b">
            <div class="l">
                <div class="r">
                    <div class="bl">
                        <div class="br">
                            <div class="tl">
                                <div class="tr">
                                    <div class="pad">

                                        <?php while( have_posts() ) : the_post(); ?>

                                        <?php 
                                        $category = get_post_meta( get_the_ID(), 'category', true); 
                                        $img_style="style="width:606;height:34;position: absolute; padding-top:1px; padding-left: 2px; z-index:9999; background: no-repeat;"";
                                        ?>

                                        <div class="post" id="post-<?php the_ID(); ?>">

                                            <h1><?php the_title(); ?></h1>
                                            <img src="https://wordpress.stackexchange.com/questions/13209/wp-content/themes/DD4L/images/leafhr.png" <?php? echo $img_style;?>>
                                            <br />

                                            <div class="entry">
                                                <?php the_content(); ?>
                                                <?php wp_link_pages(array('before' => 'Pages: ', 'next_or_number' => 'number')); ?>
                                            </div>

                                            <?php edit_post_link('Edit this entry.', '<p>', '</p>'); ?>

                                        </div>

                                        <?php endwhile; ?>

                                        <br /><br />

                                        <?php if( !empty( $category ) ) : ?>

                                        <?php
                                        $args = array(
                                            'orderby' => 'title',
                                            'order' => 'asc',
                                            'nopaging' => true,
                                            'ignore_sticky_posts' => true,
                                            'tax_query' => array(
                                                array(
                                                    'taxonomy' => 'category',
                                                    'terms' => array( $category ),
                                                    'field' => 'slug'
                                                )
                                            )
                                        );
                                        $category_query = new WP_Query( $args );
                                        ?>

                                        <?php if( $category_query->have_posts() ) :  ?>

                                        <?php
                                        $total = $category_query->post_count;
                                        $quart = $total / 4;

                                        if( floor( $quart ) != $quart )
                                            $quart = ceil( $quart );
                                        $counter = 0;
                                        ?>

                                        <div class="float-container">
                                            <ul class="alignleft">
                                            <?php while( $category_query->have_posts() ) : $category_query->the_post(); $counter++; ?>

                                                <li>
                                                    <a href="https://wordpress.stackexchange.com/questions/13209/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a>
                                                </li>

                                            <?php if( $quart == $counter ) : $counter = 0; ?>
                                            </ul>

                                            <ul class="alignleft">
                                            <?php endif; ?>

                                            <?php endwhile; ?>
                                            </ul>

                                            <div class="clear"></div>
                                        </div>

                                        <?php endif; ?>

                                        <?php endif; ?>

                                    </div>
                                    <div class="clear"></div>

                                </div>
                            </div>
                        </div>
                    </div>
                </div>
            </div>
        </div>
    </div>
</div>

<?php get_sidebar(); ?>
<?php get_footer(); ?>

See edit revision for previous comments and code.