Horrocks-Mumford surfaces are cut out in ${\mathbb P}^4$ by 3 quintic and 15 sextic polynomials; I am not sure how "explicit" and "simple" one can make these, though representation theory will help somewhat. The equations will have many dependencies (syzygies) between them. The references I found are [Manolache, Syzygies of abelian surfaces..., J für die reine und angewandte Mathematik 384, 180-191. Theorem 1] and [Aure et al, Syzygies of abelian and bielliptic surfaces..., https://arxiv.org/abs/alg-geom/9606013, Corollary 3.3]. Both of these papers contain plenty of representation theory. As explained in the introduction of the latter paper (and clear in many ways), finding an abelian surface in low-dimensional projective space is rather an "accident" and the equations are not going to be "simple". There are more "natural" families in higher-dimensional spaces, where the equations organise somewhat better; see 
[Gross and Popescu, Equations of (1,d)-polarized Abelian Surfaces, https://arxiv.org/abs/alg-geom/9606013] and references therein.