Algorithm to test whether a language is context-free
By Rice's theorem, to see if the language accepted by a Turing machine has any non-trivial property (here: being context free) is not decidable. So you would have to restrict the power of your recognizing machinery (or description) to make it not Turing complete to hope for an answer.

For some language descriptions the answer is trivial: If it is by regular expressions, it is regular, thus context free. If it is by context free grammars, ditto.