It doesn't appear that there was a formal "commissioning" of any sort. Most of the information that follows is actually summarized from articles on Wikipedia (on Louis VII of France, Bernard of Clairvaux, and Pope Eugene III).
The story apparently begins in 1144, when Louis VII of France was successfully fighting Theobald II of Champagne. At the very end of the war, Louis personally participated in burning the town of Vitry-les-Francois—including the church building, where at least a thousand people had sought refuge.
In atonement for this, and to look after some of the Franks in the Holy Land who were in danger from the Seljuk Turks, Louis decided that he wanted to mount a crusade. After Odessa fell to the Turks, Pope Eugene (Eugenius) III wrote a papal bull to him, Quantum Praedecessores, giving official papal blessing to anyone who wanted to accompany him on the crusade.
Bernard of Clairvaux got involved because he was a friend of the pope, and in fact had influenced him to enter the Cistercian order. It would seem natural that the pope would turn to him to preach on behalf of the crusade—Bernard was known as a fantastically eloquent preacher.
In short, the pope didn't formally "commission" Bernard to preach the crusade. He asked his good friend who was very well known as a preacher.