You can use either. If you place it at the beginning of the sentence, you might think about adding a comma after "morning", but it is not essential.
Neither is better or worse than the other, and there is very little difference in emphasis. However, if you don't usually leave home at 8 o'clock, so this morning was different from usual, it may be a little better to place "this morning" at the beginning, to provide some emphasis. This isn't essential, particularly not in speech, where intonation can provide far more emphasis than word order ever can.