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. Thethe 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, which organises matters somewhat. 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.
Computer algebra allows one to make explicit (if not simple) calculations. The particular case of Horrocks-Mumford surfaces is treated by macaulay2 in [Eisenbud et al, Computations in algebraic geometry with Macaulay 2, Example 7.2.].