I tried to moon-walk as soon as I got into Captain Falcon competitively and needless to say, it wasn't happening. I went at it for a little bit, a few times a week or so, and ended up just putting it on my "to do" list so I could focus on other essential tech. I would try to do it randomly during training, and after about 3 or 4 months I was able to MW to the right across FD. I've only started to be able to MW consistently to the left across FD after about a year of playing (again, this was just from practicing it a little randomly).
Moonwalking is all about dashing in a direction and then, within your dash dance range, quickly bringing the analog stick to the opposite side without bringing it through the neutral or down position. Doing a half circle is the cleanest way to do this, but it takes too much time to precisely make that your thumb do that motion. Good moonwalks come from fast analog stick movement. I've heard a lot of Falcons dashing and then just making a single diagonal line to the opposite side going in between neutral and down. Check out your controller and you'll be able to see this position easily - it's got its own notch on the octagon. This can work out well, especially if you're doing it fast enough. Personally I do the reverse Nike swoosh method, where the tail is the start, you go diagonally thru neutral and down, and then curve up at the end. I can get some nice healthy MWs doing this, but it can be tricky when doing multiple MWs since it's easy to go thru neutral when coming back for a 2nd MW.
What controller your using can also help or hurt you when trying to moonwalk. I started out using a brand new white controller bought off Amazon, and the analog stick was really tight. I bought an older purple controller that was in great condition, and the sticks allowed for easy fluid movement, but still reverted back to neutral quickly. I was able to MW easier using this controller. I've since broken in the white controller and can MW fine with it - breaking them in just takes some time. If your sticks are too loose though (like floppy 'n ****; move a lot when just barely running your thumb over it), they can really hurt your tech in general. There are some mods you can do to your controller to fix this (putting some stuff in the stick box {glue or plastic}, or replacing it with another controller's), but a lot of people just get other controllers.