You get four out of the five pieces from one quest at level 45, and the bodypiece from the final quest at 50 after you beat Curious George. There's a small walkthrough on Wiki here

http://ffxiv.gamerescape.com/wiki/Looking_the_Part

But to save you the time and trouble the pieces can be found in.

-Natalan, just to the right before the third gate (There will be two gates, it will be close to the one on the right, and you should see it on the mini-map)
-14,15 near Turning Leaf (You will need to walk off chocoback there to get by the Treants)
-Cranepearch Tower in Lower Las Noseca (On the Cape far south of Ceaderwood I believe it was, you'll probably need to walk off chocoback to avoid aggro from drakes)
-Cutters Cry just before the Princess room. You can get there if you kill the little ants last in the first room or two, then Sentries in the next few rooms last.