We present a generalization to genus 2 of the probabilistic algorithm in Sutherland~\cite{Sutherland} for computing Hilbert class polynomial˻s. The improvement over the algorithm presented in~\cite{BGL} for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for ''going up'' to a {\it maximal} curve (a curve with maximal endomorphism ring), once we find {\it any} curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of $(\ell,\ell)$-isogenies to compute all isogenous maximal curves from an initial one. This article is an extended version of the version published at ANTS~X.