Understanding how to decide whether a language for a DFA is decidable
Let me rephrase the question:


  Is there an algorithm that, given a DFA, decides whether its language is $\{0,1\}^*$?


Such algorithms do exist, hence the corresponding set is decidable.

I'm not sure what your instructor means by "high level description of a Turing machine", but what they should have meant is "describe an algorithm that solves the problem". Hopefully you already know how to do that.