Part 1: What is a proof?
There are three types of proof I will be addressing.
- Proof by admission
- Proof by deduction
- Proof by empirical inference
The first category is what happens when a defendant pleads guilty to a crime. The work of the prosecution is immediately done, and there is nothing left to argue. I therefore should not have to prove a point when my opponents are already in agreement. So if I can show that theists themselves are already claiming that "God" is not a real thing, then I don't really have anything to argue about.
The second category is what we typically associate with a logical or mathematical proof. Proof in this category relates to the idea of God as a coherent concept. If I can show that the definition of the word "God" is either incoherent or inconsistent, then it cannot possibly relate to a real thing in objective reality. The idea itself is just meaningless.
The third category is what we would typically associate with a scientific proof or a courtroom trial. There are no logical absolutes in this realm, but there are certainly reasonable conclusions one can derive from a preponderance of empirical evidence. So if I can show that the evidence simply does not justify the existence of any God, then there is no reason to suppose such a thing exists in the first place.
Part 2: Proof by Admission
Start by pointing out the existence of many hundreds of religious denominations. Show that they all believe in mutually inconsistent God concepts. Any assertion of the existence of one is automatically an implied rejection of the others. Therefore, Christians themselves are claiming that the Muslim God and the Mormon God are false views. Muslims claim that the trinitarian God is false. Mormons and Jews reject the Christians and the Muslims. All of them reject the existence of Thor or Vishnu.
The atheist agrees with all of them. None of those gods exist. We just go one step further to include your god. But it's important to realize that the claim "God X does not exist" is perfectly normal. Theists themselves are the ones making it.
Move on to the hypocrisy of religious believers. Christians claim to follow Jesus, but habitually pick and choose the what rules to follow. I've never seen a Christian "turn the other cheek" on a meaningful level. I've never seen one sell all his goods and donate it to charity. If Christians can't be bothered to actually live by the rules they profess to believe, then they obviously don't really believe in the rules or consequences. Again, proof by admission.
Part 3: Proof by deduction
Acknowledge the "weakness" of the first proof. It doesn't really disprove God, exactly, but only our agreement that such a thing isn't real. So fine, let's move on.
Begin with the definitional properties of God: spaceless, timeless, immaterial. These words are logically equivalent to the statement that God "is nowhere, does nothing, and isn't made of anything." They literally "mean nothing" in that a being like God is indistinguishable from nothing at all.
Move on to the apparent self-contradictions in God. Omnipotence is not really a thing. It contains paradoxes. Combine omnipotence with omniscience. More paradoxes. Move on to the contradictions in things like the Bible - all things are possible with God, but God cannot lie? Contradiction! Talk about the trinitarian God - God is one God. Only really God is three Gods. Gibberish!
Talk about the plan of salvation - God sent Himself to die for us so that He could forgive us after making us the way we are. Huh?
Hell - infinite punishment for finite crimes. But God is still benevolent. What?
Part 4: Inference to the Best Explanation
Assume "God" is a logically well-defined concept. Now what?
Nothing. You've just successfully defined a word. Any idiot can do that. The set of all possible worlds is infinitely greater than the set of all actual worlds. Why should I assume that just because you defined a fancy word that there automatically is a tangible referent for that thing in objective reality?
Obviously, I shouldn't, and it would be stupid to just take people at their words over this. That's the null hypothesis in a nutshell. The burden of proof is on theists to show that this word actually refers to something real. I am under no obligations to scour every last nook and cranny of physical reality to disprove such a thing. Otherwise, prove to me there are no flying spaghetti monsters.
Now take it even further. What predictions does the idea of God make? God is a maximally powerful benevolent agent. That implies a being that would minimize suffering in our world. Yet apparently, the world is full of pointless, gratuitous suffering. Human efforts consistently make the world better off than it is without the apparent help of any divine interventions. These observations are empirically inconsistent with the expectations of God's existence. The idea of God is therefore scientifically falsified.
Christians do have a whole slew of impressive sounding arguments for God, and they claim that evidence backs them up. But they don't. The arguments are always, without exception, either fallacious, incoherent, or non-falsifiable. That does not satisfy the burden of proof. A thousand bad arguments does not magically translate into one good argument. We need a theory that predicatively explains things on an empirical level. but the idea of God simply doesn't "do" anything.
Finally, we know where religion comes from. We know what psychological factors drive it. We have structured scientific theories for how religion spreads and how it manipulates believers. There is simply no need to invoke God for a phenomenon that would have happened anyway without Him. Science is in the process of explaining religion itself.