Using Cv For Non Ideal Gasses
We usually don't use the real gas $C_V(T,p)$ directly in calculating the change in U in going from thermodynamics equilibrium state A to thermodynamic equilibrium state B, if both these states are beyond the ideal gas region.  Starting from state A, what we do first is use the real-gas p-v-T equation to determine the change in U in going from the initial pressure $p_A$ to a low pressure (or large specific volume) in the ideal gas region $p^*$ at constant temperature $T_A$.  Then, our second step is to hold the pressure constant at pressure $p^*$ and raise the temperature from $T_A$ to $T_B$, using the ideal gas heat capacity to determine the change in internal energy for this step (since we are now in the ideal gas region).  Finally, we hold the temperature constant at $T_B$ and raise the pressure from $p^*$ to $p_B$ using the real-gas p-v-T equation again to determine the change in U.  The sum of the changes in U for these three steps adds up to the change in U in going from $(T_A,p_A)$ to $(T_B,p_B)$.