Decidability of directed strongly connected graphs
Yes, the DFS is an algorithm, and hence has an equivalent turing machine. The definition of the decidability as you have said, is for the language to be exactly accepted by some turing machine.
The formal definition of the turing machine accepting the language of strongly connected graphs probably is waaaay too hard to explicitly write, so I guess your teacher meant for you to show an algorithm and state that an algorithm is equivalent to a turing machine.
Btw, as a side note, you can run DFS only twice in total instead of $n$ times - once per node. If you have free time, its a good exercise in algorithms :)