You've got the right start. Let's back up a bit on that velocity equation.

So we see that the constant is the velocity of the object at t = 0. (We could have gone straight to this result, but I feel this explanation is more "transparent.")

Once you have v0 you can integrate again:

and get

and get your height.

By the way, there is a much faster way to do this, but it's a more "Physicsy" way of doing things:

-Dan