'다음 주' and '그 다음 주' are little different.
You may understand when seeing the examples.
you ask to the doctor for reserving your next visit,
and the doctor says, 다음 주에요. Visit next week.
3월 첫째주에 와도 될까요? 제가 다음 주에는 바빠서요.
Oh, can I come the first week of March? I'm busy next week.
그러면 그 다음 주에 오세요. Then, please come the next week (of the first week of March. = it means that you should come the second week of March).
So "그 다음주" can used the next week of the certain time that you set as a standard or a criterion.
"the week after next week" = '다다음 주' or '다음다음 주'.
Either way is okay and both are officially correct.
Literally, '다다음' is 'the right next(or after) to the next one' and '다음다음' means 'the second turn from this turn'. So both are similar in meaning.