"neither... nor"

Technically, the sentence

You will not be allowed to do neither the non-graded nor the graded assignment of week one unless you complete this activity

could be considered correct as is, but it would have a rather unexpected meaning. It would mean: if you complete the activity, you'll be allowed to do neither the non-graded nor graded assignment; if you don't complete it, you will have to do at least one of them.

However, based on the obvious intended meaning (i.e. if you complete the activity, you'll be allowed to do one or both of the assignments; if you don't, you won't be allowed to do either of them), it should be either:

You will not be allowed to do either the non-graded or the graded assignment of week one unless you complete this activity

or

You will be allowed to do neither the non-graded nor the graded assignment of week one unless you complete this activity

Unless you want to sound very formal, the first of these two versions sounds better ("You will not be allowed to do either the non-graded or the graded assignment...").

In some dialects of English and in very colloquial usage, the original sentence could in fact express your intended meaning (cf "you ain't seen nothing yet") - but such usage would be considered non-standard and would be strenuously avoided in this kind of relatively formal context (the rules for assessment).