My Mom did this with the sandy spot in her yard under oak trees. If nothing is growing there now, great! Just put some organic matter (OM,) mulch, leaves, lawnmower bag contents (as long as you're sure your grass doesn't have seeds on it, although it sounds like it won't grow under there anyway, so not as much of a concern as if you were doing that on a sunny bed.) Some kind of outline, to keep the OM and lawn separate would help, bricks, landscape timbers, rocks (though the irregularity is tougher to maintain,) whatever you want to look at, can afford, find for free, etc... Like you said, the boundary was already drawn around the drip line. While she was gone one morning I was over there and decided the last of the brick pile *still* sitting there from when the house was built, would be better used as a border out there, and just sat them down along the already-drawn border. The shade has continued to kill more grass, but when I first did it, it looked like it had been a bed forever, just naked of any kind of mulch or leaf cover or any smaller plants.
At least yearly, you'll get a new layer of leaves to cover the area. Anything else you can add periodically will help. The tree roots will always be greedy about the moisture, but except for plants known to need a bog, shade plants are made to cope with that, and the layer of OM will keep things so much more moist, just like the forest floor, created pretty much the same way, by putting (and leaving that which falls there naturally) OM on the surface. The difference in digging a hole anywhere in the 'bed area' at her house now is AMAZING - usually a hand shovel will do, where the first few times we tried to dig the first year, we needed a maddock even where there were no roots. (And this is how I usually start a new bed by reclaiming a grassy spot also, smothering under OM.) As long as whatever you're putting on is fairly chunky, airy, it can be several inches thick. As it starts to hold more moisture, the decomposition will happen more quickly, so the first layer will seem like nothing is happening, then suddenly you might notice, it's almost all gone.
Decide where a path will be, larger shrubs if you want any, maybe a couple of chairs and small table for enjoying the fruits of your labors, or resting while at work. When you can see it in your minds' eye, you're ready to get going! Since there are a ton of roots, you can't dig big holes, so get small plants, let the tree roots dictate where you can/can't dig.
As long as you don't radically alter the depth of cover over the roots, or attempt to insist on digging through a ton of large roots, it should go well! Shade plants are made to go under trees, and the trees are probably ambivalent about the smaller entities, but likely greedily 'use' the added leaves they may drop in the area. Coddle plants as little as possible, they need to establish deep roots by seeking water, to be self-sufficient after the first year. Have patience, dry shade is the slowest gardening. But also the least weedy, except for your tree sprouts.
Pics of reclaiming an area from grass, and Mom's front bed. Hopefully you won't fill yours with weedy Vinca, but that's beside the point. (Mom finally agreed to let me pull the Vinca up, now that's it's an unimaginably nasty chore... but the payoff will be bringing it back to my house in a big tub to compost. If I pull it, trim it, mow it, rake it, it's my OM. Yay... although she's starting to catch on after watching it happen in her own front yard for years. I wouldn't be surprised if her yard waste piles started shrinking too. We rarely discard any. Small trimmings go directly on beds, branches devoid of tons of leaves make steaks and burgers taste GOOD.)
