(I use Photoshop CS6 exclusively, so I'm not sure if this would work with past versions. Just let me know.)
What I do first is open up the Timeline panel (Windows > Timeline), and select the first frame of the GIF. Then I hold Shift, and click on the last frame of the GIF, so that all the frames are highlighted blue/selected. Then I do the same in the layers dock; I click the bottom-most layer and then hold Shift before selecting the top-most layer so that I have both the frames and layers selected completely. Then I click on the three bars on the right of the Timeline panel and choose the 'Copy Frames' option in the dropdown menu that appears. Then I open up a new document/image (for the GIF to go on), and on the Timeline panel I go over to the three bars again. But this time I choose 'Paste Frames', and then 'Paste Over Selection'. You should have all the frames on top of the image/background layer now.