My approach to atheists has been somewhat successful, for my anyway. I can't say I've won any of them to the Lord as of yet, unfortunately, but I have successfully disarmed their arguments on several occassions.
First, though, I've generally found that the true committed atheist is robustly religious and very knowledgeable, they've generally studied various religions in depth. There are some that are mean and nasty, and others that are highly intelligent and like to put on intellectual airs as if they have the high ground, I deal better with the latter than the former.
Second, I attempt to take away the high ground and show them that the term "atheist" was first applied to Christians in Rome, who denied the deity of Caesar and the genius of Rome. Hence, they can't stand on Christian ground and claim to be an "atheist," neither will I let him maintain exclusive use of the word. How hard is it, after all, to not believe in a God you cannot prove exists? True atheism looks a self claimed deity in the face and denies his claim of deity.
Hence, this step is critical because it is necessary to demonstrate that his claim of atheism is standing upon Christian ground denying the Christian God, thus he is not an "atheist," he is merely not a monotheist. So, don't mistake the modern trend of a denial of monotheism as a claim of "atheism."
I then try to turn the tables on them and show them that modern statism is a political religion and has resurrected the old divine pagan state concept whereby the atheist is standing on Christian ground denying the existence of the Creator, not deity. Then demonstrate that a lawmaker is a god to it's subjects. I then show them that I believe in the Creator God and explain the Creator/creature distinction and then express my atheism toward the modern divine pagan state and it's claim of deity.
Next, I generally try to take it another step and explain the Gospel, that its very foundation is atheism in denying the deity of yourself, whereby you must look in the mirror and deny that he who is looking back is a god.
In reality a man cannot deny the Creator exists since He created all that is, all he can do is suppress that truth and transfer the meaning of it someplace else, ascribing deity to himself or the State and it's institutions, or concealing a pantheism or animism in his claim of atheism.
Anyway, that is how I approach it.
Member, Covenant Reformed Presbyterian Church