Luigi's Purple Coins frustrated me to no end. It's been a few months since I've played it, so I'll try to remember what I can. My general path started by going to the left-outside (the side opposite the Bill/Phil/Jill/whatever Board), then staying close to the center while moving to the far corner with the 1-up. Next, I headed through the center portion that I hadn't hit, then onto the corner with the long jumps and back to the star.
-Take the time to "explore" the level a bit if you haven't. It'll help you pick out your path for your coin run. Build up a stockpile of lives beforehand if you'd like, and
always try to get the 1-up in the far corner.
-Jump. A lot. The rotating platforms are more of a pain than the shrinking/disappearing ones. Try to avoid them as much as possible. You'll also want to leave yourself a safe path to leap through on the way back around the stage, so jumping often is the best strategy. Also keep in mind that you can stay on the green shrinking platforms pretty much until they completely disappear.
-As JCpackers05 stated, the long jump is very useful. You'll need to use it to cross the larger gaps in the level (i.e. obtaining essential coins over the purple sections).
-You
should have plenty of time, so don't be afraid to collect the Star Bits that drop from the sky. This will aid you in building/keeping your stock of lives. When I did it, I always waited to start a run until I reached the nearest multiple of 50 for the life. Star Bits carry over after dying; use it to your advantage.
Hope this helps! Good luck!
Almost forgot: keep in mind that you won't have to get every single coin in the level. If you miss some here and there, it's no big deal. You'll need to get the long strings of coins over gaps, though.