My makeup table is a desk as I've found that if sitting down at a table, I can rest my elbow on the table and apply liner much better than if I'm not able to steady my arm.
I have a table mirror that can be tilted - similar to this one: www.babyliss.eu/products/beauty/mirrors/8435e/
I adjust the mirror so I can look down into the mirror - it's at a 45 degree angle or so. That means that I don't have to close my eye completely when applying liner but that I'll be able to see what I'm doing with both eyes.
I like using a gel liner, such as the Maybelline EyeStudio Lasting Drama Gel Eyeliner 24H - it's in a little glass pot and comes with a pretty good brush.
In the beginning, I did short strokes, starting almost at the outer corner of the eye. Drawing little lines in the direction of the outer corner, connecting each new stroke with the one I had just drawn (if that makes any sense :D)
I've tried different brushes and I like the one that comes with the liner (it also works for lipstick if you don't like it for doing liner), as well as a Mac 210 and a Mac 266 (angled brush). The 266 is the one I find the hardest to do a decent liner with, but it's worth trying different brushes if you can - there are cheaper alternatives to Mac if you just want to test if a specific shape works better for you.
It does take some training to get it right :)
If you cannot get the cat eye style liner right, then maybe it's worth doing something like push liner. You can see it here:
I use a Mac 212 for this, and I use either a gel liner or a dark eyeshadow.
What this girl does with a pencil liner and a brush also looks pretty and is quite simple: