Derive Turing Machine with high level description
Its easy to check that $C$ is given in Chomsky Normal Form, because you just have to iterate through the production rules and make sure each one of them satisfies the definitions for being a CNF production.
Then, the harder part is deciding whether $C$ can produce $s$ or not.
Notice, that a CNF will derive a word with length $n$, in exactly $2n-1$ steps. (every production will either transform a variable to the letter corresponding to it, or will create one new variable)
Keeping this in mind, we can just iterate through all production chains that contain $2n-1$ steps (for $n=|s|$), and check whether we saw $s$ or not. If $s\in L(C)$, then we will see it, since $s$ can be derived with $2n-1$ steps. And if $s\notin L(C)$ we won't see it, therefore the algorithm is correct.