The question basically. I was able to find rho and theta $rho$ and $theta$ as $\int_1^5\int_0^2pi$,(2pi is the top limit), respectivley. I can't find $phi$ though and I'm assuming thats wrong. My integrand is $\rho^3\cos(\phi)\sin(\phi) \,d\rho d\theta d\phi$ Can anyone help me go into the right direction? I don't know what I did wrong.
You are so very close. Your integrand is correct, your Jacobian is correct, and your limits of integration for $\rho$ and $\theta$ are correct. The only thing you are missing is the limits for $\phi$.
To figure this part out, you need to think about how $z$ is parametrized in the spherical coordinate system. You did not explicitly specify a parametrization, but one that works and is consistent with your integrand is $$\begin{align} x &= \rho \cos \theta \sin \phi \\ y &= \rho \sin \theta \sin \phi \\ z &= \rho \cos \phi, \end{align} \tag{1}$$
in which $\rho$ is the distance of $(x,y,z)$ from the origin, $\theta$ is the counterclockwise angle of the point $(x,y,0)$ measured from the positive $x$-axis, and $\phi$ is the angle that the $(x,y,z)$ makes with the positive $z$-axis--so we have $\phi \in [0, \pi]$ and $\phi = \pi/2$ will give $z = 0$, which is the "equator."
With this in mind, it is clear that the upper hemisphere will correspond to the interval $\phi \in [0, \pi/2]$. So this is your limits of integration for $\phi$, and the desired volume is $$V = \iiint_D z \, dV = \int_{\rho=1}^5 \int_{\theta = 0}^{2\pi} \int_{\phi = 0}^{\pi/2} (\rho \cos \phi) \rho^2 \sin \phi \, d\phi \, d\theta \, d\rho, \tag{2}$$ which I leave you to evaluate.
One comment I have about this problem is that I find its phrasing imprecise. "The spheres of radiuses (sic) 1 and 5" is totally vague, since we are apparently expected to assume that these spheres are centered at the origin. Less problematic but still vague is the phrase "between the spheres...." A more precise statement would be to say something like "outside the unit sphere but inside the sphere of radius $5$ centered at the origin." Or more explicitly, one could have given a system of inequalities; e.g.,