Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space and
$W=uC_{\varphi}$ be a weighted composition operator on
$L^p(\Sigma)$ ($1\leq p<\infty$), defined by $W:f\mapsto u.(f\circ
\varphi)$, where $\varphi: X\rightarrow X$ is a measurable
transformation and $u$ is a weight function on $X$. In this paper,
we study the hypercyclicity of $W$ in terms of $u$, by using the
Radon-Nikodym derivatives and the conditional expectations. First,
it is shown that if $\varphi$ is a periodic nonsingular
transformation, then $W$ cannot be hypercyclic. The necessary
conditions for the hypercyclicity of $W$ are then given in terms of
the Radon-Nikodym derivatives provided that $\varphi$ is
non-singular and finitely non-mixing. For the sufficient conditions,
we also require that $\varphi$ is normal.
The weakly mixing
and topologically mixing concepts are also studied for $W$.
Moreover, under some specific conditions we establish the
subspace-hypercyclicity of the adjoint
operator $W^*$ with respect to the Hilbert subspace $L^2(\mathcal{A})$.
Finally, in order to illustrate the results some examples are
given.