There are many ways... I found mine:
I use one of the black dripline connectors that have barbs on both ends.
I cut it in half (carefully, with a sharp knife), so one is actually good for two bottles!!!
I drill a hole into the bottle cap that is just a little bit smaller than the black thingy (so I can push it in with a little effort)
I add a little adhesive glue to the black thingy, and push it into the hole (with some pliers, or pushing the cap against something sturdy). The glue is pushed out on the sides and provides a tight seal.
Then I let it dry for 12 hours. Okay no I don't, but maybe it would be a good idea.
So now you have the bottle cap with the barbed thingy... time to push the airline over it and screw it onto the bottle.
Works every time. I tried Rex' method using only the airline, but wasn't that successful.