In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [8] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results