Not sure if Bethany and Carver count for the achievement so I will not be including them in this guide. Also I'm only doing how to get their friendship and not rivalry. Although I suppose for rivalry you could just do the opposite of what I say to do.
General note: Do your companions side quests as it will help a great deal.
Aveline - I used most of the "good" responses, top right-hand corner". When I had no option of the basic responses I tried to use the response for what I thought would be the nice thing to say. For missions I tended to notice that she disapproved of criminal type missions or during missions when I was being unfair.
Varric - During conversations with him I usually used the funny/joking response and got his friendship quite easily.
Isabela - I was always nice to her and did all her side quests, gave her a gift "ship in a bottle", and got her friendship easily as well. She also approves of you asking for money for your quests.
Merrill - Simply was always nice and comforting to her. Friendship came soon after
Fenris - One of the hardest friendships for me because I usually support mages. Fenris will approve of you not siding with mages so go the templar route. During the Bait and Switch mission I would advise against bringing a mage on the second half. Note: Fenris is a hard person to read on the conversations at his house so I would advise to save before you talk to him so you get the proper outcome.
Anders - Was pretty easy for me because I went with the mages. So basically support mages all the way and you'll get him easily. Also be nice to him.
Sebastian - I do not know much on how to get his friendship although I got it in the end so I will tell you how I dealt with him. When I brought him along I saved a lot to make sure I got his approval, did his companion quests and did the right or "just" thing to do.
That wraps up my very first guide, hope it helps you guys. If I left anyone out I'm sorry and please let me know and I'll try to fix it if possible.
Edit:
"I can confirm one of two things. Either you only have to have 4 companions max out as friends and rivals, or your siblings count and you need to have the 4 of the same. I maxed them out in this order: Bethany (friend), Varric (friend), Fenris (Rival), Anders (friend), Merrill (friend). It popped for me right after I maxed out Merrill." - Quoted from boxandthefuzz
So either I was wrong about siblings, if so you have my apologies, or you can get it from conflicting sides. More reports on how you got the achievement would help us get an idea on what works and what doesn't. Thank you.