In the interest of giving an answer for an old question:
Try taking a normalization of your curve, and then studying what happens to Euler characteristics and degrees when you pull back line bundles to the normalization. Each of these two questions have simple answers.
Then upstairs on your new smooth curve you can apply the old Riemann-Roch formula, and end up with the formula you want!
(Hint: "pulling back" commutes nicely with "twisting by divisors," so the only line bundle you ever really need to pull back is the structure sheaf. By definition, it pulls back to the structure sheaf, so degrees don't change. Now what is the difference between the Euler characteristics of the two structure sheaves?)