None of us are perfect human beings. You made mistakes and all you can do at this point is ask to talk to him in person. Look him in the face when you apologize and face towards him. That's a start.
First, accept responsibility for your actions by admitting what you did (e.g., "I went out with John behind your back and covered it up by lying to you"). Don't add qualifiers or try to explain it away (e.g., I was feeling lonely because of you ... ") because this is about how YOU own what you did. Attempts to explain it away will make matters worse at this point. Make sure you offer a heartfelt apology.
Second, describe how your behavior impacted him, how it made him feel. This lets him know that you truly get what you did to him.
Third, tell him what he means to you and ask how you can begin to make this right. If he is not to start forgiving you, then tell him you'd like to talk more with him in another few days to see if he is open to working with you to repair the broken trust in your relationship.