Basically what i did here is correct some loose ends that you had on the sprite itself (AA, better handling of the shadows and highlights)
Then i took an edit on the animation itself.
Your biggest allie when animating are mirrors; Even with something as basic as walking (that everyone is supossed to know how it works)...is to watch yourself do it; observe what moves, what comes first, the pacing and what happens when you do it.
Then sketch it out; draw the rough key frames, some preffer silhouttes; fill the rest of the frames. Does it make sense? do you understand with it being a sillhoutte what its supossed to be doing? if the answer is yes, keep going, if not, you need to review
After you feel good with it; flesh it out! put the colours, decide how the lightsource and shadows will vary, what kind of character is? a walk can show you the personality of the char; is it confidente, goofy, sneaky? it all depends on the animation.
Pacing and timing is very important; when you walk for instance (and that was one of the things wrong with your walk) is that many things happen at the same time; Not only you move one leg at a time, the other stays behind for a split second, knees bend, feet rises your overall height decreases at some point and you recover it at others.
Timing is giving that keyframe more or less emphasys; some movements are more long in time than others, others are affected by the starting velocity, or recoil; all of this you must take in account.
I also added more frames; You had 4 and now it has 8; though it can work pretty much flawlessly removing 2 transition frames; you can also make a good animation in 4 good keyframes, the problem with your animation is that you didnt use them good; you only moved one leg at a time, and the other stays in the same position; like i said, get a mirror and watch yourself; Does your other leg stay in the same position?
Observe, you are no longer looking, but watching, how it works, how it moves.
I gave your character a particular personality, try to study what i did there, and apply it once more with a brand new walk, lets see what you can do!
-Keep pushing it, this isnt a talent, is a logic thing, if you try and persevere, youll get better, is as simple as that.
-Krut