I don't understand why you're using the notation f(x0), f(x1), f'(x0), f'(x1). I'm going to assume they're just constants.
It's just a matter of doing the algebra.
p(x)=a0 + a1(x-x0) + a2(x-x0)^2 + a3(x-x0)^3
p'(x)=a1 + 2a2(x-x0) + 3a3(x-x0)^2
As you've already noticed, a0 is easy, since p(x0)=a0. Also a1 is easy, since p'(x0)=a1.
So you have
p(x1)=a0 + a1(x1-x0) + a2(x1-x0)^2 + a3(x1-x0)^3
p'(x1)=a1 + 2a2(x1-x0) + 3a3(x1-x0)^2
and since you know p(x1), p'(x1), a0, a1, x0, and x1, you have two equations in the two unknowns a2 and a3.
If you're still having trouble, please post again in this thread.