If your question is: can you present the homeotopy group in terms of generators and relations, the answer is "yes", following the work of Hatcher-Thurston, Wajnryb, and most recently M. Korkmaz, who gives a relatively civilized presentation. If you mean: given a homeomorphism, can you express its isotopy class in terms of the generators, I assume that the answer is yes, but it obviously depends on how the homeomorphism is given. For related work, see
Brinkmann, Peter(1-UT)
An implementation of the Bestvina-Handel algorithm for surface homeomorphisms. (English summary)
Experiment. Math. 9 (2000), no. 2, 235–240.