How can I get over the past in my relationship?

My boyfriend of five months expresses how much he cares for me and loves me, and I believe it. It has been one of the best relationships I have been in for a short period of time, but he has lied to me so much. He'll tell me that he'll stop lying, and while he's saying that, he's hiding something or lying about something. First, it was about a girl. He hid text messages and claimed she was a friend and it was a friendly conversation when it was a girl he once found interest in. Next was his ex-girlfriend. He put a fake contact name, and it was just friendly, but he lied to my face. All the time, he claims he never cheated and still expresses his love the same way and that he wants to change and do better.