In the past I've sent off homebrews to national competitions and larger bottles of other potent potables to friends and family, so I looked into all that stuff. It's been a few years, though, so not sure how the laws may have changed.
First off, do NOT use the USPS. 90% because it's very illegal, 10% because no matter how you label the package it's still got a good chance to get the crap beat out of it. So, spend a little more and go with FedEx or UPS.
Still, both of them won't ship fragile liquids, but not sure anymore whether alcohol's actually illegal to ship via private carrier or just against those business' Terms and Conditions, which are much less a hassle to break. So, you lie. At least with liquor you won't have to invent zany reasons for them to always keep This End Up (homebrew sediment issues).
Line the shipping container's insides with plastic, as best as you can. If there is breakage, you want to both (i) minimize the collateral damage and (ii) not get caught. Shoeboxes always worked great for me. To prevent breakage, use an oversized container and stuff it stuff it stuff it with packing. Not peanuts or leftover amazon bubbles; go with good old-fashioned wadded-up newspaper (who says print is dead?). It's a great combination of solid and cushion, with no hard edges and little risk of over-compression. Also a good sound-proofer, so no sloshing sounds. And really stuff it in there so it's tight, zero wiggle. Then seal the bottle itself as waterproof as you can, say if you have a large enough ziploc-type freezer bag. Otherwise, a trashcan bag and lots of tape. Go ahead and add a layer of bubble wrap before cramming it into the box of newspaper, and add more newspaper to the lid side til it's not so easy to shut. Pack it well enough, and you may even get away with not asking for a Fragile sticker, which comes with its own questions (and more lies).
Over-tape all edges and seams, wrap that box in plastic again, taping it sealed, then use good old brown package paper from USPS to sloppily wrap it all up to look like any other ol' package with your handwritten to/froms on it.
It all may sound excessive, but I never went to jail or paid a fine, so, paranoia and over-cautiousness does get the job done. Take half-measures at your own risk.