In this paper we develop an approach to Bayesian Monte Carlo inference for skewed α-stable distributions. Based on a series representation of the stable law in terms of infinite summations of random Poisson process arrival times, our framework leads to a simple representation in terms of conditionally Gaussian distributions for certain latent variables. Inference can therefore be carried out straightforwardly using techniques such as auxiliary variables versions of Markov chain Monte Carlo (MCMC) methods. The Poisson series representation (PSR) is further extended to practical application by introducing an approximation of the series residual terms based on exact moment calculations. Simulations illustrate the proposed framework applied to skewed α-stable simulated and real-world data, successfully estimating the distribution parameter values and being consistent with other (non-Bayesian) approaches. The methods are highly suitable for incorporation into hierarchical Bayesian models, and in this case the conditionally Gaussian structure of our model will lead to very efficient computations compared to other approaches.