What is the expiation for one who swore by Allah not to do something, and then did it, and will Allah forgive him?

If what you have resolved to abandon is unlawful (haram), then sincere repentance from it is obligatory, along with a firm resolve not to return to it in the future. As for your statement, "I bear Allah witness that I will not do such-and-such," if you intended it as an oath, then you must offer expiation for the oath (kaffarat al-yamin) if you break it. However, if you did not intend it as an oath, then no expiation is required of you.