In my opinion, neither is correct.
The phrase last week refers to a stated time in the past. In this case you should use the past simple tense.
Example: I was busy last week.
The present perfect with for is used to refer to a period of time, so you could say:
I’ve been busy for a week.
You can also use the present perfect with a period which is not over: I’ve been busy this week(The week is not over, for example it is Saturday today).
You can read more about the present perfect and do some exercises here