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.
