Mass Effect: Legendary Edition (PC) - Achievement - Paramour I
Establish a romantic relationship in Mass Effect 1, 2 or 3
*SPOILER ALERT*
If you're playing the games in chronological order, the earliest opportunity to achieve this is towards the end of the first game, just before going to Ilos. It does require that you've shown interest in at least one potential relationship up until that point.
If you're playing as male Shepard, your options are limited to Ashley or Liara. (So don't be an ass to the one you're going for) In my playthrough, Ashley died on Virmire and I picked Liara throughout anyway as Ashley was coming across as a bit of a God-loving space-racist, which I never really noticed when I played the original over a decade ago.
If you show interest in both, there is a point around midway through the game where they both come to speak to you and try and force you to pick one of them. You can only establish a relationship with one, so make your choice. If you do pick Ashley, make sure you stick with her on Virmire - This will cause Kaiden to be killed instead of Ashley.
If you are playing as female Shepard, your choices are Kaiden or Liara (again), but the gist is the same - Just be nice the the one your most interested in eventually it will come into fruition. There is nothing special you need to do other than generally have positive conversations with them.