Suppose $X$ is a smooth variety over $\mathbb{C}$. Let $K^{b}(X)$ be the homotopy category of bounded complex of coherent sheaves, and $D^{b}(X))$ be the derived category of bounded complex of coherent sheaves. One can define: $$Hom^{\cdot}(A^{\cdot}, B^{\cdot}): K^{b}(X) \to K(Ab)$$ as $$Hom^{i}(A^{\cdot},B^{\cdot}):= \oplus Hom(A^k,B^{k+i}),$$ $$d(f):=d_B \circ f - (-1)^{i} f \circ d_A.$$ Then we can pass to the derived categories, and define the right derived functor: $$RHom^{\cdot}(A^{\cdot},): D^{b}(X ) \to D(Ab).$$

My questions is: How to compute $RHom(F,k(x))$ in the derived category? Here $F$ is a sheaf on $X$(viewed as a complex in $D(X)$ concentrated in degree $0$), and $k(x)$ is a skyscraper sheaf on $x \in X$. In particular how to compute $RHom(k(x),k(x))$?

I think by definition, one has to do an injective resolution to $k(x)$, and perform the computation in the homotopy category.