The obvious answer is "don't," but that's not particularly helpful.
I don't mean to discourage my Christian brethren, who are called by their religion to preach the Good News to everyone, from attempting to fulfill their great commission as Christians. But let's be honest. Someone who has declared for the atheist* squad, in our hyperreligious American society, in which our political leaders must end every speech with a reference to God, is probably (a) pretty aware of the basic Christian proposition, and (b) not going to be convinced by any reference you're likely to advance. The tools you need to be effective are just not in your toolbag.
* - In this entry, when I use the term "atheist," I am referring to people who reject the concept of spiritual religion. Technically speaking, Buddhism is an atheist religion, but I am not including Buddhists in the term. I might just as well have referred to "secular humanists" or "freethinkers."
(I assume, of course, that you are genuinely interested in actually convincing your target to believe. If your goal is merely to expose, it really doesn't matter what tactics you use. You can stop reading now.)
Most of the atheists I know--which is a secretly large number--know Christianity better than many Christians. They've read the Bible, and maybe even studied it. Many of them are former Christians who saw too many contradictions in the religion.
The fundamental problem is that atheists simply don't accept what Christians must take as their first article of faith: that God exists and is knowable through the Bible. Whether or not you accept the Bible as the infallible word of God, Christianity requires at least some belief that at least some of what the Bible says is true. There is very little that all Christians agree upon, but that bit is universal.
That is something that atheists simply cannot accept as a first principle. Atheists believe in what can be observed and deduced and proven. If faith is an essential step in a belief, atheists will not accept it.
Some people suggest that atheists should at least make Pascal's wager--choosing to believe on the chance that the Bible is right, on the basis that making the wrong "bet" will mean the loss of everything. The atheist turns that wager around and suggests that if the Bible is wrong, and there is nothing after death, and this life is all we have, then making the wrong "bet" means the loss of everything anyway. It's not a winning argument.
So, how can Christians witness effectively to atheists?
It doesn't start with the Bible. It doesn't take words at all--in fact, words are hurtful to the process, not helpful.
Well, it sort of starts with the Bible. In John 13, the story of the Last Supper is told. Just after Jesus tells Judas Iscariot to go and betray him, Jesus acknowledges that his time on earth is drawing to a close. At verse 34, he says, "A new command I give you: Love one another. As I have loved you, so you must love one another."
Then, at verse 35, "By this everyone will know that you are my disciples, if you love one another."
Very few of the atheists I have known wish to tread on others' sincerely held religious beliefs. But virtually all of the atheists I have known despise hypocrisy. They look at Christians who speak the language of hate, of scorn, of separation and division and condemnation, and they repay it in kind. They see Christians who post all sorts of prayers, or statements about prayers, on social media sites, which are the modern-day streetcorners, and they see Christians who cannot follow even the most basic tenets of their religion. They see Christians who preach prosperity theology and drive expensive cars and live in expensive houses and send their children to expensive private schools, and they shake their heads.
Christianity is supposed to be a religion founded on love, and there is precious little Christian love evident to the average atheist.
So if you want to witness to an atheist, it begins and ends with acting like a follower of Christ, focusing your conduct on love, so that everyone will know of your Christianity, just as Jesus said in John 13:35--not because you said it (because you didn't say it), or because you talked about Jesus to someone (because you didn't), or because you prayed on Facebook (since that's the modern-day streetcorner (Matt. 6:5)), but because you lived the philosophy that Jesus espoused.
You may never manage to convert an atheist, and that's fine. Very few atheists are interested in, or even open to, a religion that requires faith in the unseen (Heb. 11:1) and discourages reasoned inquiry (Prov. 3:5). But virtually all of them are interested in the existence of better Christians--and by "better Christians" I mean "people who are better at following Christ's example" and not "people who are good at scaring people into submission" or "people who like to criticize others' faults." Come to think of it, pretty much everyone would be better off with more of those around.