Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $π$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehatπ_k)_{k\ge 1}$ of probability measures which approximates $π$ as $k$ growths to infinity. The probability measure $\widehatπ_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehatπ_k)_{k\ge 1}$ to $π$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $π$ is known.