This paper develops systematically stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process $A$ such that $[N,A] = 0$, for any continuous local martingale $N$. In particular, given a function $u:[0,T] \times \R \rightarrow \R$, which is of class $C^{0,1}$ (or sometimes less), we provide a chain rule type expansion for $X_t=u(t,X_t)$ which stands in applications for a chain It\^o type rule.