Pull-ups (palms facing out) are among the most difficult, functional exercises you can do. Quality, not quantity, is key.
Make it Hard
Use a wide grip. Before you get to the top, pause and move your head to the right as close to your fist as possible. Do the same to the left, then raise yourself over the bar.
Rather than pull up to your neck, pull up to your lower chest; in one motion lift your body above the bar. Do a dip.