The new style faucet connector cap pictured below is much more sturdy that the old style with the little pip on it.  The old style cap is still useful for lower pressure applications (such as gravity fed setups), but the new style cap provides a much more robust way of connecting the tube.
The connect the cap to the tube, feed the tube through the hole in the tube.  If you are struggling to get it in, cut the tube at an angle so you can get a start, then use long-nose plyers to pull the tube through.  It should be a fairly snug fit.
Using a straight connector barbed fitting to insert into the end of the tube as shown below.  The fitting will make the tube wider at the end as it goes over the fitting, and that will make it impossible to go back out through the hole.  Pull the tube back and give it a firm tug to seat it nicely in the hole of the cap.  Being made of polyurethane, as the tube jams in the hole it will make a very god seal.  You don't absolutely have to but you can trim the other end of the fitting off so that you can still use a strainer if you need one.
A strainer is only generally needed if your water supply contains any entrained grit.