I know that c =1. Easy enough to do. I don't know how to do the rest though.
OK, if you insist. Here it comes, ready or not...
You could try $\displaystyle f_{X+Y}(z)=\frac{\mathrm d}{\mathrm dz}\mathrm P(X+Y\leq z)$, where perhaps $\displaystyle \mathrm P(X+Y\leq z)=\int_{X+Y\leq z}f_{X,Y}\mathrm dA=\int_{-\infty}^\infty\int_{-\infty}^{z-y}f_{X,Y}(x,y)\mathrm dx\mathrm dy$.
So maybe $\displaystyle f_{X+Y}(z)=\int_{-\infty}^\infty f_{X,Y}(z-y,y)\mathrm dy$.
Tired now, need sleep... $\displaystyle f(zzzzzzzzzzzzzzzzzzzzzzzzzzzzz$