Dependence of U with respect to V
Let's calculate $\left(\frac{\partial U}{\partial V}\right)_T$ using the Gibbs equation $dU=TdS-PdV$.
We divide the Gibbs equation by dV keeping the temperature constant :
\begin{equation} \left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial S}{\partial V}\right)_T-P \end{equation}
Using Maxwell's relation, $\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V$ , we get:
\begin{equation} \left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial P}{\partial T}\right)_V- P=\frac{\alpha T}{\kappa}-P \end{equation}
In the above equation we used the relation: $\left(\frac{\partial P}{\partial T}\right)_V=\frac{\alpha}{\kappa}$
Dependence of U with respect to T
Keeping the volume constant, the heat exchanged coincides with the change in internal energy and therefore:
\begin{equation} \left(\frac{\partial U}{\partial T}\right)_V =C_V \end{equation}