The definition of the coefficients $a_{r,k}$ is given by theorem III.17 on page 53 of "Elliptic curves in cryptography" by I.F. Blake, G. Seroussi, N.P. Smart (Cambridge University Press, 1999).
With $v,s,l$ as in the question, define $f(\tau) = \left( \dfrac {\eta(\tau)} {\eta(l\tau)} \right)^{2s}$, where $\eta$ is Dedekind's $\eta$ function. If $j$ is the invariant function (see page 47 of the quoted book), then there exist numbers $a_{r,k} \in \mathbb Z$ such that
$$\sum _{r=0} ^{l+1} \sum _{k=0} ^v a_{r,k} f(\tau)^r j(lt)^k =0 \ .$$
Now just formally replace $f(\tau)$ with $x$ and $j(l\tau)$ with $y$ to get $G_l(x,y)$.