As you play the game, you will be given certain dialogue options that have a heart as part of the dialogue for your companions. Keep selecting the heart of the person you want to be in a relationship and eventually, they will ask you to do a quest for them. Once you complete the quest, the companion will ask if you really want to be in the relationship. You are given several chances to pick either to be with them or not but once you choose that person, you should just select the heart icon each time to stay in the relationship. Eventually, this will lead to your relationship blossoming and a special cutscene will play out. The achievement will pop after that cutscene. I've left an example below.