Q: When talking to an unbeliever about Christ, is it necessary to first prove that God exists?
In the first place, the Word of God presupposes that God exists and begins on that premise (Gen 1:1). It therefore cannot be wrong for the child of God to do the same.
Secondly, unbelievers may doubt the existence of God, but only a comparative few will be downright philosophical atheists. In fact, experience teaches us, and the Bible affirms, that there are no natural atheists in this world. Man is created in the image of God and, despite the Fall, retains some knowledge of the existence of God. This is why man has a conscience which no irrational creatures have (cf. Rom 2:14). This is also why, no human society, no matter how primitive or advanced, is without some form of religion (cf. Rom 1:20–21). In other words, atheists (i.e., philosophical, rather than practical, atheists) are made, not born. Attempts to prove the existence of God before proclaiming Christ may, therefore, turn out to be more distracting than beneficial. Nevertheless, we should take note that intellectual atheists are generally unwilling to listen to the Gospel; and also that those with higher (godless and evolutionalistic) education, who are willing to listen to what we have to say, may sometimes need to be reminded that God exists and will judge us for our sins.
But thirdly, remember that the message of apologetics is insufficient for salvation, while the message of the Gospel is absolutely necessary. Thus, it is often prudent to disregard the possibility that our listeners may be atheistically inclined and go straight to presenting the Gospel. Faith and belief in the existence of God are inextricably tied together (Heb 11:6). If God regenerates the sinner, his doubts concerning the existence of God vanishes instantly.
Having said this, it must be admitted that there are occasionally situations in which highlighting the evidences for the existence of God, before speaking about sin and the remedy in Christ, may be helpful. Such is the case if the listener violently objects to the existence of God and therefore shuts his ear to what you have to say. In my personal experience, a case in point would be intellectual scholars from China who have been systematically brain-washed by their communistic education system. In such cases, demonstrating the existence of God may help the person to listen more attentively and with a less biased attitude. However, the listener must be shown the unreasonableness of atheism, and also be informed that atheism is sin.