$f_{XY}(x,y)=3x~~0 \leq y \leq x \leq 1$

$F_U(u) = Pr[U < u] = Pr[X-Y<u] = Pr[X-u<Y]$

$F_U(u)=\int_0^1 \int_{\max(x-u,0)}^x f_{XY}(x,y)~dy

~ dx = \int_0^1 \int_{\max(x-u,0)}^x 3x ~dy ~dx$ for $0 \leq u \leq 1$

$f_U(u)=\frac{\partial}{\partial u}F_U(u)$

this should get you started