Is it possible to remove variable repetition here?
This class looks like a utility class, and you could use static functions: you then wouldn't need to instantiate it:

class StringHelperK {

    public static function normalizeStringK($string) {
        return static::removeAccents(
            static::replaceSpacesWithSlashes($string)
        );
    }

    public static function removeAccents($string){
        return strtr($string,'àáâãäçèéêëìíîïñòóôõöùúûüýÿÀÁÂÃÄÇÈÉÊËÌÍÎÏÑÒÓÔÕÖÙÚÛÜÝ','aaaaaceeeeiiiinooooouuuuyyAAAAACEEEEIIIINOOOOOUUUUY');
    }

    public static function replaceSpacesWithSlashes($string) {
        return strtolower(str_replace(" ", "-",$string));
    }
}


To normalize a string, you would then do: StringHelperK::normalizeStringK("blabla").

And no, $string should definitely not be a property, and I see nothing wrong with passing is as an argument to each function.