How do you talk to your non-believing family about God?

Look for natural openings during conversations. Also be aware of their input, its a good sign of interest. Understand these things can take time so don't push to hard. Take it one step at a time.

Be very, very careful to avoid "lecturing" or using a patronising tone that you have found the secret to eternal happiness and they haven't.
Be honest about your testimony if they want to know where you stand. But you have to respect their non-compliance and atheistic bent. it's their choice. take it or leave it.

If you exhibit an authentic change, they'll be asking you about it. You won't have to figure out how to broach the subject with them.

As stated before, don't be pushy. Don't be preachy, and don't act "holier than thou".

Jesus4Ever said:

How do you talk to your non-believing family about God? If you witnessed to your family, how did you do it?

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sometimes, as written, (for believing spouses for their unbelieving spouse) ,
without a word - as written, without a word, the other(s) (family, friends, neighbors, co-workers) MAY be saved without a word IF they see a completely changed , new, different way of life.

Never shrink back to destruction though, also as written....

As others have said, don't be preachy. If your family is like mine, just the mention of God, will cause a huge fight. I decided that we, as a family, couldn't discuss God. I would instead try and be an example. If they asked, I'd give some testimony, but I accepted too, they weren't ever going to ask. All I could do was pray. If your family is open, share, but don't get preachy. If they aren't ready to hear it, then be an example and don't push it on them. It only makes things worse.

You can try a few times to share some things, but if they push back and don't want to listen, then fall back on praying for them, because you have done what you could.
When it is time to go, then make a clean break of it.

Matthew 10:14
And if anyone will not receive you or listen to your words, shake off the dust from your feet when you leave that house or town.
Mark 6:11
And if any place will not receive you and they will not listen to you, when you leave, shake off the dust that is on your feet as a testimony against them.”
Luke 9:5
And wherever they do not receive you, when you leave that town shake off the dust from your feet as a testimony against them.”
Luke 10:11
‘Even the dust of your town that clings to our feet we wipe off against you. Nevertheless know this, that the kingdom of God has come near.’

Titus 3:9-11New King James Version (NKJV)
Avoid Dissension
9 But avoid foolish disputes, genealogies, contentions, and strivings about the law; for they are unprofitable and useless. 10 Reject a divisive man after the first and second admonition, 11 knowing that such a person is warped and sinning, being self-condemned.

Jesus4Ever said:

How do you talk to your non-believing family about God? If you witnessed to your family, how did you do it?

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Before you talk to an unbeliever about God, talk to God about the unbeliever. Because, the only way that anyone can accept the things of God, is for the Spirit of God to change their heart.

Do not marvel that I said to you, ‘You must be born again.’ The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.” ~ John 3:7-8