The trick to beating Lost Levels is to finding the character that works for you. If you'd like the best method, that will also help you on your way towards saving over the million coins for the "buying everything" achievement, then read on.
Pick Juni. We may have people say otherwise in the comments, but her strengths ended up being what allowed me to start beating this mode with ease. Not only can she double jump, but she can slowly glide, and also slide down and jump off walls, She will save you an incredible amount of times.
Once you get used to how she controls, I'd be shocked if you didn't beat the Lost Levels on your first try or two. Just make sure to never go out of your way for a coin. Some of them are better off left untouched and saving you a life.