With this work and its sequel, Moduli of unipotent representations II, we initiate a study of the finite dimensional algebraic representations of a unipotent group over a field of characteristic zero from the modular point of view. Let $G$ be such a group. The stack ${ℳ}_n\left(G\right)$ of all representations of dimension $n$ is badly behaved. In this first installment, we introduce a nondegeneracy condition which cuts out a substack ${ℳ}_n^{\mathrm{nd}}\left(G\right)$ which is better behaved, and, in particular, admits a coarse algebraic space, which...