Does waking up for Fajr prayer at the first call to prayer become obligatory after the intention to atone for a lapse in performing it, and is expiation required if one fails to adhere to it?

Your statement: "It has become obligatory upon me to wake up at the first call to Fajr prayer." If you intended by this a future commitment to atone for having slept through prayers, then this wording indicates obligation and commitment, and a vow is thereby established. So, if at the time of utterance you intended that you were permanently obligating yourself to this awakening, then you are liable for an expiation for a broken oath (kaffarat yamin) every time sleep overcomes you. However, if you did not intend that, then you are only liable for an expiation for a broken oath if sleep overcomes you for the first time, and there is no expiation upon you thereafter. The expiation for a broken oath is feeding ten needy persons, or clothing them, or freeing a slave. If you cannot find these, then fasting for three days.