We present a generalization to genus~2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Br{ö}ker-Gruenewald-Lauter algorithm for the genus~2 case is that we do not need to find a curve in the isogeny class whose endomorphism ring is the maximal order; rather, we present a probabilistic algorithm for ''going up'' to a maximal curve (a curve with maximal endomorphism ring), once we find 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 is an extended version of the article published at ANTS~X.