This answer varies from atheist to atheist. Some are more stringent than others.
A good place to start is following the scientific method. Science is the single most effective method humans have ever conceived for precisely and accurately determining what reality is, and how it works. It is incredibly picky, harsh, slow and difficult. It works.
The computer you're using to view this webpage, over the internet, was not prayed into existence, it was "scienced" into existence.
The problem is that most theistic claims cannot, by definition, adhere to the scientific method. Many are not falsifiable, or even testable.
If you wish to undertake this endeavor, this might be a good process:
- Define what it is you're trying to demonstrate is true, and what its attributes are.
- Derive some possible tests that would demonstrate those attributes.
- Perform those tests, gather up the evidence.
- Submit for good, rigorous peer review.
- If it survives the peer review, submit to the person you're attempting to convince.
Yes, it is difficult, however, we have many volumes of scientific knowledge that is managed to pass this process with flying colors, and has resulted in virtually all advanced technologies in the world today.
Often, theists will attempt to intimidate ("You burn in hell!"), or suade with emotion ("But Jesus has shown you so much love!") to convince an atheist. These typically don't work. We do have emotions like anyone else, and appreciate it, but when it comes to the matter of accurate truth about reality, it must come down to the logic and evidence.