In this work, we consider the problem of estimating the probability distribution, the quantile or the conditional expectation above the quantile, the so called conditional-value-at-risk (CVaR), of output quantities of complex random differential models by the Multilevel Monte Carlo (MLMC) method. We follow an approach that recasts the estimation of the above quantities to the computation of suitable parametric expectations. In this work, we present novel computable error estimators for the estimation of such quantities, which are then used to optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We demonstrate the efficiency and robustness of our adaptive continuation-MLMC in an array of numerical test cases.