Note: This will require being unfaithful to your love interest from the first game. Continuing Liara's romance in a DLC for the second game will not grant you a Paramour achievement. To minimize my personal feelings of guilt, I recommend romancing either Ashley or Kaidan (whoever you like the least) in the first game, then choosing a romance option from the second game
) to continue with into the third.
Mass Effect 3 has ten romanceable characters for the purpose of unlocking the achievement.
Female Shepard can establish a romance with Samantha. Romance with Garrus can only be rekindled from Mass Effect 2.
Male Shepard can establish a romance with Ashley and Steve. Romances with Jack, Miranda and Tali can only be rekindled from Mass Effect 2.
Both of them can establish a romance with Kaidan and Liara. Romance with Kelly can only be rekindled from Mass Effect 2.
(Five more characters can be romanced (albeit in a different, usually toned down way). However, they will not unlock the Paramour Achievement. They are James and Javik for female Shepard. Thane is also available only if continued from Mass Effect 2. Diana can be romanced by both as well as Samara but only if continued from Mass Effect 2.)
For additional information about Romances, check the wiki page on the subject as it wouldn't be sensible or useful to cover everything in this guide.
https://masseffect.fandom.com/wiki/RomanceCredit to Fauved for their
guide.