First you need to figure out what the problem is. Unless the yard was filled with mostly organic matter, or there is a serious erosion problem, it isn't very likely that it has 'sunk'. What is more likely is that the grass isn't full enough to compensate for mud season. Blindly raising the level of the backyard could just give you a bigger problem, like a wet basement.

BTW, you probably don't want topsoil. Too much of it is junk, and it's hard to get the good stuff because the professionals grab it. If you end up with that level of lawn repair, a truckload of compost will give you more bang for your buck.

Try wood chips for areas that get heavy dog traffic.