How does the partial pressure of gas C change with the change in volume of the container?
The equilibrium constant $K_P$ is independent of pressure but does depend on temperature which we assume to be constant. The equilibrium constant $K_P$ is the ratio of partial pressures $p_i$ of the gaseous substances raised to the stoichiometric powers, $K_P= p_C^4/p_B^4$ and is defined at a standard state of $1$ atm. pressure.

Writing the equilibrium constant for the reaction $\ce{mB &lt;=&gt;lC}$ in terms of the mole fraction $x_i=n_i/n$ for n total moles gives $K_x= x_C^l/x_B^m$ where l and m are constants, both four in your question. The mole fraction i of a species behaving as a perfect gas can also be written  $x_i=p_i/P$ where $p_i$ is the partial pressure and P the total pressure. 

Rewriting the equilibrium constant in terms of partial pressures of B and C gives
 $$K_x=\frac{p_C^l}{p_B^m}P^{m-l}= K_PP^{\Delta n}$$
where $\Delta n$ is the change in the number of moles of the gases. Thus unless the change in the number of moles is zero, which it is in your example, the mole fractions of the components  will change with change in total pressure even though $K_P$ is constant. If $\Delta n &gt; 0$ the pressure will reduce the mole fraction of products.