I don't know what you did or what happened to cause them to duplicate, but if you have duplicates in the finder it would stand to reason that they would duplicate in the playlists.
Click on the Music category (top left of iTunes) then choose the File Menu and select Show Duplicates. From there you can select all and delete (Command Delete) all of the duplicates, and move the files to the trash when prompted. This will clean the duplicates out of the iTunes folder in the Finder as well as remove them from the iTunes Library file. You can also hold down Option before selecting the File Menu and View Duplicates will change to View Exact Duplicates. Exact Duplicates may do a better job of differentiating between a studio and live version of the same song for example.
There is a Show All button at the bottom of the duplicate search to get you back to a regular view of your iTunes library.
I would recommend making a backup of entire ~/Music/iTunes folder before messing with this further.
There are certainly other ways to handle this if that doesn't look like it will give good results. I'd need a bit more info though. Are the duplicates in the same directories as their original songs or not. You can tell this by right clicking and choosing show in finder for a pair of them. Does that open two folders, or are they in the same folder, and one of the files has a space and the number 1 at the end of the file name?