I think there are two main things to keep in mind.
1. Choosing to disclose is a much more powerful position than being caught keeping a secret.
This means that if you have reason to know or expect that your employer will run a criminal background check, you absolutely want to get out in front of it and disclose before you are "found out." If it means they drop you from the short list, you probably had no chance of getting the job anyway and should just move on. If they don't drop you right away, it transforms the background check from something that "outs" you to something that reinforces your honesty (provided you were completely honest, and didn't try to play off aggravated assault as "a little scuffle," for example).
Since disclosure can earn you honesty points, and you never really know for sure if a background check will be run or not, it might be a good idea to always disclose as a rule. This is your choice, and as for the timing, I'd say near the end of your first interview would be appropriate. By that point, they've formed an impression of you; if they don't like you, it doesn't really matter either way, and if they do like you they'll be a little more receptive.
Why not earlier?
For instance, why not disclose at the beginning of the interview? In your cover letter? At the bottom of your application?
It's a general rule that people remember the first and last part of an interaction better than anything in the middle. This disclosure doesn't really belong in your cover letter; it'll color their view of you before you even meet. Similarly, don't drop a bomb right away when you first shake hands; both you and they have already committed to sitting down and talking for anywhere from a few minutes to the rest of the day, so make the most of it. Don't let it be their first impression; wait until you've covered most of the "meat" of the interview, and if you feel like you've shown you are qualified and have good character, then you have reason to hope they'll see the disclosure as a reinforcement of that. Having some questions ready to ask them afterward, perhaps as a way of highlighting your interest and professionalism, might be a good way to make sure it's not their last impression, either.
Just keep in mind how much time you've both already invested and how much more you're willing to invest without that disclosure on the table.
2. You are under no obligation to volunteer information that's not relevant to the job.
If you're not comfortable with "always disclose" then the question becomes, what was the crime, and what is the job? Depending on the nature of both, it may be more or less important to disclose right away. For example, if you're applying for a job as an accountant and you were convicted of felony embezzlement, you need to disclose right away to avoid wasting everybody's time. It's directly relevant to the job and could be a deal-breaker for most employers.
On the other hand, let's say you violated a custody agreement and ended up with a felony kidnapping conviction (I don't know how likely this is, but it is possible). That conviction probably isn't relevant to most jobs that don't involve children. I wouldn't expect an applicant to necessarily volunteer that information if it's not relevant and they're not asked. Some employers will feel differently, but people often view family law issues as inherently private matters, and if an employer really wants to know about any and all felonies, they can often ask about it right on the application.
The only thing I would say is invariable is that if you're asked about your record, even very indirectly, at any point in the process (including once you are hired), you should answer honestly, erring on the side of complete disclosure. It could be a question as innocent as, "Is there anything else you'd like to tell us about yourself?" at the end of a phone screening. You probably can't afford to take the risk that they don't want to know at that point, as it would be reasonable for them to feel misled when they find out later.