You don't go into detail about how direct or obvious you have been in showing interest in your violin teacher, but assuming you have said things to her which a married man ought not, there is not going to be an easy way of breaking this to her. Not without her thinking you've been playing her, ironically, like a fiddle.
Your stated goal is to put this right and keep your marriage, so you need to remember that whatever fallout you get from this, your marriage is your priority. What you are doing, you are doing for your wife and your family stability. If you get some grief, lose the friendship, lose your violin lessons, or wind up looking like a bit of a jerk, that is just what you're going to have to go through for the sake of your marriage. Take the hit, and learn the lesson.
Personally, I do not think it would be a good idea to continue lessons with this tutor given that you admit feelings for her. In fact, once you tell her that you are married it might make an affair easier because you might learn that she is prepared to sneak around behind your wife's back and that will just make the temptation more real. If you really want to save your marriage, removing the temptation would be the wisest course of action.
If you only see the tutor weekly for lessons and have no dialogue outside this, then frankly you don't really owe her an explanation. You could just cancel your lessons and find a new tutor. This may not sound like much of an interpersonal solution, but communication is relative to each situation and here the situation is you have been paying her for violin lessons. You are a customer, client, student, whatever you want to call it, it is a business arrangement and you could end it in a businesslike fashion.
If you really believe you can overcome that temptation, continue lessons and have a friendship, then the best approach would be to take the initiative to tell your tutor straight the next time you see her. As you have stated, this is better than waiting for the subject to come up 'naturally', such as the next time conversation gets flirty. You need to "own" the mistake you have made, not pretend everything is fine and act like you just "forgot to mention" your wife. So walk in there, take control of the situation and say what you got to say.
If you go with this option I also think that you might be safer, and even claw back a bit of your self-respect, if you just tell it from your perspective and not assume any feelings on her part. Even if she does have feelings for you, if you have never actually confirmed this or done anything about it, there is a good chance that when you tell her you are married she will put her defences up and deny as much as she can. So just make this about you, about putting things right that you have been worried about.
Adapting what you have said in your question background, you could perhaps say:
Before we begin the lesson, I just want to speak to you about something. This has been bothering my conscience for some time.
These violin lessons have been time taken out of my regular life, something for me, and because of that, I haven't told you any great detail about my life, or about my family. I never told you that I was married, my wife's name is [name], we've been married for 10 years. I've been feeling guilty about never mentioning her to you, and I just wanted to clear that up.
Using your wife's name is important - it will keep this priority clear in your mind, and make your situation clear, and more real to your tutor. You have been deliberately obscure about your life, it is time to change that around and be completely transparent. Also, whatever you do, don't admit feelings for her, even if she admits them to you. It won't make her feel any better about the situation, it can only make her think that there is a chance of a relationship, or that you are in conflict. Resolve to neutralise the situation you have created.
I can't guarantee her reaction to this, but it is at least direct, and gets it over and done with.
