Pass the category id into `get_category_link()`:

    <?php
    $category = get_the_category();
    $link = get_category_link( $category[0]->term_id );
    ?>

**Update** Outputting in template:

    <?php
    $category = get_the_category();
    $first_category = $category[0];
    echo sprintf( '<a href="%s">%s</a>', get_category_link( $first_category ), $first_category->name );
    ?>