Am I able to get a Schengen visa using a 1951 Convention refugee travel document issued to me by Ukraine?
Can you apply for a Schengen visa? Yes.
Will you get a Schengen visa? Probably not, unless you can show very strong ties to Ukraine, such as a highly-paid job, a piece of real estate, marriage to a local citizen, etc. Considering the latest refugee-related events it is most likely hard to convince the authorities you will actually leave at the end of your stay.