In this paper, we are interested in the extinction time of continuous state branching processes with competition in a Lévy random environment. In particular we prove, under the so-called Grey's condition together with the assumption that the Lévy random environment does not drift towards infinity, that for any starting point the process gets extinct in finite time a.s. Moreover if we replace the condition on the Lévy random environment by a technical integrability condition on the competition mechanism, then the process also gets extinct in finite time a.s. and it comes down from infinity under the condition that the negative jumps associated to the environment are driven by a compound Poisson process. Then the logistic case in a Brownian random environment is treated. Our arguments are base on a Lamperti-type representation where the driven process turns out to be a perturbed Feller diffusion by an independent spectrally positive Lévy process. If the independent random perturbation is a subordinator then the process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case and following a similar approach to Lambert [11], we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Riccati differential equation.