This page contains all the information you need on how to have a romance with Javik.
Javik is a romance option available to the Female Shepard. Before you can begin the romance with Javik, you’ll need to complete the mission, Eden Prime to find and recruit him to the team.
Once he is in your team, you’ll want to make sure you’re not in any active relationship and then wait until the end of the party during the Citadel DLC mission. Here you’ll watch a scene where you wake up in Javiks bed. There is nothing else to this romance and it is never mentioned again.