If there is no soul or self, why did the Buddha speak of his past lives? How is it possible that he had past lives and how did he know they were his? What defines them as his?
How is it possible that he had past lives?


Because until he became the Buddha he hadn't eliminated craving.
Craving leads to clinging, clinging leads to becoming, becoming leads to birth.


  how did he know they were his? What defines them as his?


Do you remember your childhood? If so, why do you call it your childhood? Because your current existence(assuming you are an adult now) is conditioned by how the child behaved. In the same manner, if your mind had the power to go further back in time, even before your birth, you would see what actions you performed in your last life to condition your present birth. In that sense, it makes it your past life.

But here we are merely talking about a causes and effect relationship. Such a relationship does not require an existence of a soul. A mango seed does not require a soul to result into a mango tree. It rather requires sunlight, water, fertilizer, air etc. Water does not require a soul to turn into steam. It requires heat. 

In order for something to qualify as a soul, it has to be under your total control. Which means, it shouldn't be existing without your consent, it shouldn't be changing without your consent, it shouldn't be dying without your consent. Can you find anything in your mind or body that meets those conditions? Could the child have stayed as a child if he/she didn't like to become an adult?