Suppose we are over an algebraically closed fieldof characteristic zero, and $G$ is connected. Then, since we have an exact sequence $$1\to U\to G\to G_r\to 1,$$ where $U$ is the unipotent radical of $G$, and $G_r$ is a reductive group. Since a semisimple or unipotent group is generated by unipotent elements, by the Levi decomposition theorem, this implies that $G^u$ is the intersection of the kernels of all the characters of $G$. Characters of $G$ are grouplike elements of the Hopf algebra ${\mathcal O}(G)$. So the additional relations are that some grouplike elements $g_j\in {\mathcal O}(G)$ generating the group of characters of $G$ are equal to $1$.
Suppose we are over an algebraically closed field of characteristic zero, and $G$ is connected. Then, since a semisimple or unipotent group is generated by unipotent elements, by the Levi decomposition theorem, $G^u$ is the intersection of the kernels of all the characters of $G$. Characters of $G$ are grouplike elements of the Hopf algebra ${\mathcal O}(G)$. So the additional relations are that some grouplike elements $g_j\in {\mathcal O}(G)$ generating the group of characters of $G$ are equal to $1$.