Rabbi Nachman of Bratslav's advice for how to pray is good advice for how to pray the penitential psalms and supplications during this midnight watch (and all during the Days of Awe services) so that we don't fall into rote recitation but let the words carry us beyond ourselves:
Rabbi Nachman admonished us strongly to put all our strength into the words of prayer.
He said that a person must force himself a great deal when he prays. A minority opinion holds that a person shouldn't force himself in prayer. But this is not right--a person must force himself with all his strength when he prays.
Rabbi Nachman also said that when a person prays with feeling--that is, when he connects his thoughts to the words, paying attention and listening to what he is saying--his strength is automatically drawn into the words of prayer. This is because a person's strength automatically waits and looks to be drawn into holy words.
When a person prays with feeling, all his powers are drawn into his prayer. Then he prays with great strength, even though he isn't forcing himself.
(The Chambers of the Palace: Teachings of Rabbi Nachman of Bratslav, ed. Y. David Shulman, 119-120)