Grass growing help from an expert?

I'm trying to get some grass seed for my backyard, I have had failed attempts for getting grass to grow here in the past. My backyard is somewhat of a jungle. Lots of shade, and tall trees surround the top of my hill in my backyard. My yard starts flat coming off my back deck then about 6 ft away goes uphill to a steep incline. I can not get grass to grow for nothing in my backyard. It is a mud pit. I was thinking about digging a trench so rain will flow away from my yard that will solve the seed washing away. But my yard only gets a few hours of sunlight a day what seed could I spread back there? I plan on rented a rototiller machine, and ripping up whats back there which is just a bunch of moss and some stubborn weeds and starting from scratch. Then I figure I would need topsoil and hay. Anybody know some specific good quick grass growing tips? I want it to grow quick and outlast the shade and rain, also whats the time frame for some good grass? :) I do have a dog, should I keep him away from my back yard while its growing? (If these are stupid questions...sorry I'm a rookie grass grower)