Dirichlet's proof is described in <A HREF="https://books.google.nl/books?id=qEwpwWyVPIAC">Number Theory:  Algebraic Numbers and Functions</A> (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. 
<A HREF="https://home.mathematik.uni-freiburg.de/arithgeom/lehre/ss20/algzt/Keith%20Conrad%20-%20Unit%20theorem.pdf">[source]</A>

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 <A HREF="https://mathoverflow.net/q/106196/11260">in this MO question</A>.