I noticed that its really hard to find actual suggestions here. I wonder if it has more to do with the fact that people don't really fish much when backpacking because they are moving fast, or because they don't want to share secrets... I have heard the AT from northern NH through ME is just amazing for flyishing. I ran into some guys who were picking up rods in Lincoln, NH for the rest of their journey north.
The key is to really get back there. I second the idea with the Northville-placid trail as a great location for fishing. I've fished a lot in areas surrounding, but not as remote as the ponds and lakes around that trail and have seen incredible numbers of trout. For practical advice, I suggest you think about planning your trip around the fishing - you'll discover that during July and August fish are not as active and you won't have a way to fish your wet flies in slow/placid ponds. Also casting becomes difficult because you have these very dense forests and few to no clearings along the shores of all these little rivers and ponds. Others are in swamp-like areas, offering very little casting opportunity. I ALWAYS wish I had a pack raft when I'm fishing areas like these... you need to get in the middle of the water to really access it. Terrain is not going to be friendly and its quite the opposite from the terrain out west and above tree line. Using waders and/or hip waders is going to help a lot. Using tenkara is going to be much better than traditional fly fishing. Often spin casting is useless as these fish don't care for spinners and the like. Worms are also often difficult to find along these routes... but somehow work the best. If you want results, fishing wet flies/nymphs and worms under the dams made by beavers or near places where the water flows and is oxygenated will yield the best results.
In terms of areas in the traditional backpacking regions of the northeast like Adirondacks and Whites, you tend to see the fish down low and not at altitude. There are really no fish or just really tiny fish in the high mountain rivers. Taking very small tenkara and tanago poles will greatly increase your results. I often take my daiwa soyukoze rod with me and have caught quite a few 4-6 inch fish in places that don't look like there are fish... but more times I have found that there is just nothing there - and that the reality is there are no big fish. Dry fly fishing is hard but if you are very lucky you want to find places with placid surfaces and use very small dries. I like flies that can do both... the "ausable ugly" and also the ugly bug type flies are awesome. My favorite is a variation called "the usual" which works well for me pretty consistantly in the ADK mountains and much of the rest of the northeast.
My conclusion is that overall backpacking and fishing are somewhat at odds with eachother and that this is why there isn't more information readily available to us about good locations. Much of the time when you are really back there you are moving fast and not stopping to relax and enjoy the time you would spend fishing. Waking up extra early to fish, or planning your hike to end at a fishing location at the right time isn't always easy. Most of us don't have the option to just stop for a day or two extra on the way to take advantage of a great fishing spot.
All this being said, I suggest carrying a tenkara rod while backpacking and just trying out the spots you find... otherwise you'll have to accept that you are really fishing and that the hiking is secondary - then pick a great location and just go there to fish!
Now to get specific - The Ausable River and its tributaries, The Beaverkill River and its tributaries, the north fork of the boquet river, and the Delaware River water gap and tributaries are all winning spots in New York. In the ADK west region, you have the oswegatchie, the Indian River, the upper Hudson River, and Cedar River flow. The white mtns in NH have some great options too - The Saco River and its tributaries in particular are great. Then the southwestern lakes region is just full of small ponds and lakes that kick ass and need major exploring. Overall, get ready to get dirty and get wet to find the native fish, and don't expect them to be huge.
Edit to add more info.