I think the phrase "I am having a time of my life" means that a person is having a moment for himself/herself. This could be in a way of reflecting about something or perhaps having a moment of freedom and solitude. This is what I understand about this but you can correct me if I'm wrong.
The other way to put this into perspective is by phrasing it as: I am having the time of my life. This phrase now means that a person is having a very memorable moment of his or her life--something that he/she would consider one of the best. For instance, when a person tells you, "I hope you had the time of your life." It basically means that he/she is hoping that you have enjoyed whatever it is that you did and that you consider that as one of the best experiences you've ever had.