This one is a pain in the ass! I had to do a second play through to get it as it's easily missed! Make sure to follow this as you go. So as soon as you meet her...
The first comment you can make is say "it's exciting to be in the rover" no bonus is given though
-After doing the Excavation mission she'll ask you to look into her father's death, this is important as it's the start of your relationship and her side mission.
-Go to the doctor Scott, chat with him and then grab the requested notes. Bring them back to Amelia and there's a few options about the information, just tell her the news.
-Now you'll go to meet Scott's ex-wife, she's nearby and there's a few options to get the information. Collect it whatever is easiest for you depending on class builds and inform Amelia once more.
-Complment her after, I used "don't talk like the rest of them" which worked fine though no difference or negative relationship status
-Apologize to Amelia during the rover drive (story related, yet an option that comes up)
-Have her as you companion when you leave Ophir, talk to her about "private matters" when you have time. Ask about "romantic relationships" and "her interests" for +4 relationship, do not ask about "her childhood" ever and now she will be "friendly with granted bonuses.
-When the rover is back give her 100 serum for tools and help find the parts. Let her fix the Rover, she'll ask you to step aside afterwards to find details about her father.
-Talk to Phobos about Scott's son (located right beside the Rover), this well send you to the Ophir Archives. Find the archives and you'll gain +3 on relationship.
-Talk personal matters with her once complete, compliment her and mention "her travels" for the best results as this will gain +4 on the relationship.
-Later on around Mutant City she'll ask to talk to you alone, you'll need to go find her father's Rover with her. There you'll find a tablet with what happened, give her the tablet. Convince not to do the action, I used charisma to do so.
-Go to the Captain, you'll most likely use her for support at that point anyways and the second part of this Rover quest. You'll need to wait for a few in-games hours as the Captain finds research. It's best to sit there and go do something as you need to fight enemies to get back to her. After the time is up and the information is revealed it'll add +3 on the relationship. You can talk to her again right after, Amelia says next time you're at your room she'll pay you a visit.
-Find a bed, kiss her and then choose to stay in the relationship. The achievement will then unlock and you will be "in love" with the character.