To complete Home Sweet Home you will need to talk to Shank and establish 3 Raider outposts in the Commonwealth. (It is advised that you pick the smallest outpost first and after you claim it start prepping for 100% happiness by building beds inside of shelter, food, water, defenses, nuka radio transmitter and a booze distillery.)
After staking your claim you will need to make sure your Raiders stay happy as mentioned above. As a rule of thumb I went 1 extra for every settler I had IE: If I had 3 settlers I built 4 food/beds/water and 2 turrets, by the time I got to this part since I set up ahead of time I already had 100 happiness and could continue on.