Loc: Gateway to Columbia Gorge
I looked up the El Yunque National Forest in Puerto Rico. It's the only tropical rain forest in the US system. Unfortunately, the "recreation" page on the website has been replaced by a page about the horrendous damage caused by last year's double-whammy hurricanes.
I haven't been backpacking in the Carribean myself, but I'd be curious to hear about a trip if you do one. As Mouse mentioned, Dominica is known for its long trail. I did a little reading and actually went so far as to become a follower of the trail's Facebook page so as to keep track of their recovery from the hurricane, thinking I might go someday. According to their posts, which I honestly don't pay much attention to, they have been working hard getting back up, and maybe it is ready to be visited. Whether its beauty has recovered may be another matter. The place I think might also have a lot pf potential is the Dominican Republic (similar name, different island). Home to the tallest mountain in the Carribean at 10,000'+ and some good scenery, I have read. Backpacking? Don't know. When you do your searches, search for "trekking" so as to avoid the confusion of hostel travel hits. Most of the world uses the term "backpacking" differently than we do, referring instead to low budget travel using hostels, and such. A good search for info can be the Thorntree Forum, run by Lonely Planet Books. Have fun!