Adding custom field and querying from post table

Adding a column to the posts table is a bad idea, this is indeed what Custom Fields (Post Meta) are for. Query and sort your posts based on the values of your custom field with WP_Query and a post meta (custom field) query:

$query = new WP_Query(
    array(
        'meta_key' => 'counter',
        'meta_value' => 0,
        'meta_compare' => '>'
    )
);