Samurai is very cute, I like the ideas behind the animations. What I would adjust is:
1) Run: Make his left arm bob less (like his right arm), spread the feet out from his center point a little so they don't touch as much. With how cutesy his design is, you can probably show the whole underside of each foot wen it swings up. Also animate a hair bob.
2) Punch: Very cute anim. When he turns to the side before striking, make his left foot less of a full sideview - it's a bit jarring. The position his left foot ends up in is also a bit up and high. Try moving it down 1 pixel and in 1-2 pixels. I'd get rid of the swoop art on the foot, as well as his left fist, leaving it only on the right fist. A little trick if you add even more UMPH! to the punch - in the frame where it freezes and is big now, replace that first with one even bigger (maybe 1 pixel out on all sides), then on the next frame pull it back to the fist you have now. That'll give it a springy, super-strong swing, enough that the fist actually recoils a little.
3) Double punch: Get rid of the swooshes (too much for a weaker punch) and spin the body a little with each punch. Give him a unique pose per fist, even if the differences between each pose are minimal. This whill make the whole thing feel snappier.
Cheers,
Adam