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. 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. [source]
There is an oft-repeated story that the idea for the proof came to Dirichlet while he was attending Easter mass in the Sistine Chapel in Rome. Attempts to document that story are described in this MO question.
