This quest, assigned by Captain Kells, becomes available after completing Duty or Dishonor and after starting Liberty Reprimed. Kells has some unfinished business with Virgil in The Glowing Sea. Simply put, Kells wants Virgil eliminated.
How Virgil will react when you see him depends what you've done with The Institute. If you destroyed the Institute without giving Virgil the serum that would have cured him (see The Glowing Sea), he will want to kill you, making killing him an easy decision. If you did cure him, you can still kill him or let him go.
If you opted for the latter, you will have to lie to Captain Kells when you give your report. Whatever you decide, talking to Kells will be your last step in this quest.