I believe your question raises a few other important topics and questions which you may want to consider when defining the success factors of such a society, so I will split my answer up accordingly:
If your citizens live in a modern world, particularly a world with cheap, ubiquitous and uncensored global communication (e.g., the internet), your success at establishing and strengthening your society will depend on your society's cultural influence (a.k.a. trans-cultural diffusion), more so than any laws you put in place to curtail the open observance of religion.
What kind of government do you run?
You've mentioned you will allow religious observance in private homes, so presumably you would not require any sort of Orwellian Thought Police (generally regarded as a bad career move, politically speaking!)
Is your society a country? (Given you intend to global trade, immigration, and set broadly-scoped laws that would impinge the constitutional rights of many current world nations, this seems likely.) If so, is it a democracy? Or do you spread your society's beliefs via dictatorial rule and military might? (Unfortunately numerous precedents exist even in modern world history.) If you are democratic, then the need to "sell" your anti-religious beliefs to your people becomes as important (if not more important) than convincing outsiders.
How your society (and its leader) came to power is another equally important topic. Did a large number of like-minded people get together and create a cultural movement large enough to earn real power and influence (and presumably land) in the world? Or did a small number of people use fear and/or brute force to carve out a bastion of power defiantly standing against all the bad, nasty, evil religions in the world? ("Infidels!")
Can we come up with a better name than "post-religion"?
What's in a name? perhaps, but given that you seem to want to (and need to) sell your ideals to your populace (and, likely, the world, if you do not want your borders and population to shrink over time), a good name might go a long way. "Post-religion" does have some interesting connotations (c.f. post-secularism), but even with some very creative propaganda, pundits in your world will likely be quick to point out that your country is, in fact, anti-religion (as Cyrus has insightfully already mentioned).
Perhaps that kind of tension is precisely what you want to create, but if it isn't, think hard about the image your society wants to project. Often, being for something is easier than being against something else. Would a "secular society" or a "scientific nation" be better received by your world community?
How far would it go?
A self-proclaimed "non-denominational" school in Calgary, Canada was recently fined for refusing to let students pray on campus. While your fictional country would presumably be free from legal and civil actions such as this, the article I've linked does highlight some of the practical problems with attempting to keep religious practice out of public, out of schools, the workplace, etc. What lengths would your citizens have to go to in order to observe their religions? What lengths would your government go in order to stop them?
If your society is extreme enough, it wouldn't be hard to imagine the ultimate consequence of this to be a holy war, which is pretty much the definition of one group enforcing the idea that their beliefs (or in this case, the vehement lack thereof) on someone else.
Hopefully the above talking points will help you (fictionally) mold this society according to the story you want to tell.
Importantly, it does seem from your question as though your society wants to spread the influence of your "one true God". The fact that your society's "God" is an affirmative lack of religion doesn't actually change the equation that much from a practical point of view.
My advice would be, to consider what kind of socio-political parameters your "religion" will have: For example, some religions are welcoming, some are not. Some are tolerant, some are not. Some reject science, some do not. Then, look for real-world religions with similar parameters (regardless of actual beliefs), and look at how adherents have historically spread those religions, and how those religions market themselves today.