Thanks for writing! In your area, the best time to start a lawn is in late summer or early autumn, though spring is also fine.
One note, though: Grass rarely grows well in a shaded area, especially when there's competition from tree roots. Instead of lawn grass for the area, I'd highly recommend shade-loving groundcovers. They'll look better and save you a ton of time and money in the long run as you only have to plant them once (no reseeding necessary) and you won't have to ever mow them. Plus, many groundcovers have attractive flowers or foliage, so they can be a great way to add more interest to your yard!