Add content to /wp-admin/plugin-install.php admin screens

Without extending the class, what can be done is adding a Custom Action Link:

add_filter( 'plugin_install_action_links', 'action_links_wpse_119218', 10, 2 );

function action_links_wpse_119218( $links, $plugin )
{
    if( isset( $_GET['tab'] ) )
    {
        switch( $_GET['tab'] )
        {
            case 'featured':                                         
                $links['my-action'] = "Tested up to <a href="#">{$plugin['tested']}</a>";
            break;                                                   
            case 'popular':                                          
                $links['my-action'] = "Requires <a href="#">{$plugin['requires']}</a>";
            break;                                                   
            case 'new':                                              
                $links['my-action'] = "Slug <a href="#">{$plugin['slug']}</a>";
            break;                                                   
        }
    }
    return $links;
}

And then, you can proceed to move the element around with jQuery:

$('.move-me').each(function(){ /* your_magic(); */ });

---

That custom link, My view, is added with:

add_filter( 'views_plugin-install', 'views_wpse_119218' );

function views_wpse_119218( $views ){
    $views['my-view'] = '<a href="#">My view</a>';
    return $views;
}

This would be used to create your own custom screen/table with add_action( 'load-plugin-install.php', 'callback' ).