I thought I'd offer a high-tech alternative for certain varieties. If $X$ is smooth and projective over a field $k$ then <a href="http://front.math.ucdavis.edu/0204.5218">Bondal and van den Bergh</a> give a proof here that $D^b(\mathrm{Coh}X)$ is saturated which is a strong representability condition on cohomological/homological functors to the category of $k$ vector spaces. It follows immediately that $D^b(\mathrm{Coh}X)$ has a Serre functor by using the fact that $Hom(A,-)^*$ is representable for every bounded complex of coherent sheaves $A$.