Moss on lawn

I have a north facing garden and the bit of lawn nearest the house which hasn't see the sun since about November has got quite a lot of moss. 

I have been using a treatment on it to kill the moss which has worked quite well in that the moss has turned brown and I've been able to rake quite a lot of it away each week. It's really thick though and I think I'm a long way off from it all going.

Obviously, Im keen to get rid of these brown patches of moss and lay some new grass seed as soon as I can.

Should I just bite the bullet and completely dig it up?

Does anyone have any tips?

Many thanks