How to add style to category link?

You can use the_category filter to hook a callback function like this:

add_filter('the_category','add_class_to_category',10,3);

function add_class_to_category( $thelist, $separator, $parents){
    $class_to_add = 'my-category-class';
    return str_replace('<a href="', '<a class="' . $class_to_add . '" href="', $thelist);
}