Here is a computation of the quiver. I don't know how to get the relations. Let $G$ be a finite $p$-group and $K$ the $p$-element field. Note that the trivial module is the unique simple module and the radical $J$ is the augmentation ideal. It is well known that $J$ has basis the elements of the form $g-1$ with $g\in G\setminus\{1\}$. Note that

$(g-1)(h-1)= gh-1-(g-1)-(h-1)$

and so

$gh-1+J^2=(g-1)+J^2+ (h-1)+J^2$.

Thus $J/J^2$ is a $K$-vector space with generators $[g]$ with $g\in G$ and relations $[g]+[h]=[gh]$. In other words, $J/J^2$ is the universal elementary abelian $p$-group image of $G$, which is $G/\Phi(G)$ with $\Phi(G)$ the Frattini subgroup of $G$ (the intersection of all maximal subgroups $=[G,G]G^p$).

Thus $\dim J/J^2 = \log_p |G/\Phi(G)|$ and so you have one vertex and that number of loops. I would guess GAP could find a set of generators for the Frattini quotient and then your path algebra would map the loop corresponding to $x\Phi(G)$ to $x-1$ and I assume there are algorithms to get the kernel.