First off, welcome!
I agree with eishiya, and I’d like to add that YouTube has a built in slow motion feature that makes it a lot easier to analyze reference videos. You can also skip frame by frame, I think you use the “.” and “,” keys but I’m not completely sure (if all else fails, just pause and unpause really fast with spacebar or the mouse).
Aside from eishiya’s point about the legs, your animation would benefit from incorporating secondary motion. Secondary motion (or secondary action) typically refers to the motion of things that trail the “main” action. For example, clothing, hair, an animal ears/tails. The secondary motion will follow the path of the moving thing it’s attached to, but it will lag a few frames behind. So if the dog’s head is down, its ears and tail might be up, and vice versa. It’s probably easier to understand if you see it visually, so here is a handout I found with pictures relevant to your drawing.
Large image, please click here!Notice the “wave” motion - this motion is being caused by the fact that the dog moves up and down vertically as it runs. This is also an animation principle called the “wave principle”. You can practice it most easily with pencil and paper if you’d like to try it out. This is one of the first exercises they teach you in animation school, and it’s a great one, because it can be applied to lots of things.
Another large image. Click!If you want to learn more about animation principles in depth, a great book is The Animator’s Survival Kit by Richard Williams (lots of great examples used in virtually every animation class). Or you can just ask for clarification/explanations about stuff here on Pixelation, it’s definitely a lot to take in, and easier to learn if someone helps you work through it. Four-legged run cycles are one of the hardest things to do, but you have a decent start here and your game idea is very charming!