Translation:You must speak Czech as much as possible.
I am not a native Czech speaker, but I would guess that "You must speak as much Czech as possible" is not accepted because nejvíc is an adverb, making co nejvíc an adverbial phrase that modifies the verb mluvit -- therefore, “speak Czech as much as possible” would be correct. If I am wrong about this, one of the CZ natives on the team will set us straight. :-)
Sure, i completely agree, that this is an adverbial construction. But I'd argue that speaking Czech as much as possible and speaking as much Czech as possible are semantically the same and translations are that, moving the meaning from one language to another. If it can be done closely in style and grammer the better, but it's not about reconstructing the used grammer at all costs.
I would suggest that there is a difference, though perhaps somewhat subtle, between "speaking as much Czech as possible" (quantity) and "speaking Czech as much as possible" (frequency) in English. The Czech natives on the team can determine better than I can which is a more appropriate translation of the Czech sentence, or if both are acceptable.