It's hard to give advice on situations like this when so little is
known. Grass can grow nicely in partial shade, but you need to use
varieties like creeping fescue that are suited to shade. There are
some excellent shade mixes available. However, it all depends on how
much shade and the climate. For example, an area that is on the
north side of a house that gets zero direct sunlight is virtually
impossible to grow grass in. Areas that get some filtered light for at
least part of the day can support shade tolerant grass.
The tree surface roots are a major problem. Some species, eg maple
varieties, are known for having surface roots. These suck up the
moisture and nutrients and make it virtually impossible to grow grass.
You also can't use a tiller or aerator on these areas. If you're gonna
try to grow grass in shade, make sure you thin out the trees with
surface roots first. You can also thin out the tops of some of the
existing trees to get more light in.
As for killing the moss, if you're gonna seed anyway, I'd use roundup
to kill the moss/weeds, then go from there. But the bigger problem is
figuring out what the area really can support, getting the proper soil,
adjusting the PH, etc. Getting some professional advice may be the