When you witness to an atheist, it is called apologetics. Apologetics means “defence of the faith!” We have to lay a ground work with people who do not believe any god is possible, let alone the Living God!
There are different schools of thought on the best way to do this. Some people believe you should just preach the gospel right out of the Bible. And they have a point, because the word of God does not return void.
Others believe you need to tackle science, world view, answer objections before you preach the gospel. And they have a point, because you do need to reach people where they are at.
There are other schools, including both of the above. That is my preferred method. I also think it is vital to pray for this person, for God to soften their hearts.
I’ve been reading a book called “The Reason for God” by Tim Keller. He uses the first method. But I like it, because he is not talking theory. He and his wife went into Manhattan and did outreach to the young, disaffected intellectuals there. And his church grew from converts he made by both preaching the Bible in sermons, and reaching out socially.
There is also a video video series of the same name, based on the book, but handled differently. He has a discussion with a hip group of young professionals who are unsaved. He gradually answers all their objections with respect and dignity. We are watching the videos in our small group, and I am impressed by the genuine way he reaches out to these lost people, some of whom are atheists. I knew he was doing something right, but the videos show him modeling ways to develop relationships AND show his firm and honest faith in Christ.
I would suggest you might like to get the book and/or videos, to see his strategies, always with the power of the Holy Spirit to use his obvious gift of teaching people with the gospel.
Finally always remember that it is God who saves people not you! We do have to be obedient and share the gospel. But it is God who saves them. That takes a lot of pressure off the whole witnessing process, leaving you free to develop a sincere relationship with the person, while trusting that God is in control, and he will do the saving