Dirichlet's proof is described in Number Theory: Algebraic Numbers and Functions (starting on page 48).
Dirichlet did not use Minkowski’s theorem; he proved the unit theorem in 1846 while Minkowski’s theorem appeared in 1889. Dirichlet’s substitute for the convex-body theorem was the pigeonhole principle. (An account of Dirichlet’s proof in German is in [2, Sect. 183]and in English is in [6, Sect. 2.8–2.10].) Dirichlet did not state the unit theorem for all orders, but only those of the form $\mathbf{Z}[\alpha]$, since at the time these were the kinds of rings that were considered.