I really, truly believe that the key to achieving a flawless cat's eye flick at the outer corner of lined eyes is practice. Eyeliner was the one thing we could get away with wearing in school - bear in mind now that this was before anyone had heard of Amy Wine-o and her scrawled-on-with-a-felt-tip-pen liner - so I've been working on my technique for a wee while.
An applicator with a built-in brush is absolutely the best product for a beginner to start with - I can't say enough about Ruby & Millie's i-Writers or Clarins' Liquid Liners. Both have short-ish tapered brushes, which are far easier to wield than the sort of long skinny brushes of Rimmel's Glam'Eyes or Lancome's Artliners.
Don't overload your brush. You may find it helpful to rest your elbow on a table while looking down into a mirror; if you're really not confident in your lining ability, start by literally just dotting liner right along your upper lashline and then joining the dots.
I like to start close to the inner corner and work outwards, creating a straight line of even thickness all across my natural lashline. I then go back in and do the inner corner, again working from the inside out, creating a line that gradually thickens to the width of the rest of the liner so you have a nice tapered effect in the inner corner. You can take the line right into the corner of your eye and waterline or, if you're like me and inclined to paw at your peepers, have the line taper out just shy of the corner because you'll only rub it off otherwise. Once that's sorted you can work on the outer edges - just go over the original line you've created, thickening it from about the middle of the eye outwards to the corner, to get that curved line effect.
Now, for the wing. You want to angle it upwards - the sharper the angle, the more graphic the effect, so keep the angle fairly shallow if you're after a more natural look. One option is to extend the initial line that you draw out past the corner of the eye, so that you wind up with a straight line that runs out from the inner corner. Then just fill in the "V" you've created between the outer corner of your lower lashline and the end of your liner to create that wing.
Personally, I find it easier to put a little teeny dot of eyeliner where I want the end of my flick to finish and then work back in a downwards curving motion towards the lashline. A third option, which is probably more suited to someone using a pencil liner or seperate brush and gel/cream liner, involves holding a piece of card tight against the skin, angled upwards from the outer corner of the eye, as a template. (I've also seen people do this with Post Its and sticky tapes, which can't possibly be good for the delicate eye area.)
Don't expect to be able to do this perfectly the first time around, and don't try it for the first time as you're getting ready for a big night out! Do keep some cotton ear buds and eye make-up remover on hand in case you need to do a spot of correction. And do remember that, with practise, you will be able to do this on autopilot.