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 best route.