Instructions
Talk to "Kelly" during down time between missions. Ask her personal questions to learn more about her and to get her more comfortable with you.
Tell her you'd "Do more than just catch her" when she discusses the trust exercise to open up more flirtatious discussion. Keep using flirtations discussion topics in your conversations with her.
Keep flirting with her until the conversation topic of "Private dinner" opens up. Ask her to join you for a private dinner in your quarters where the two of you will become romantically involved.
Open up a secret date with Kelly by making sure to save her from the collectors during the suicide mission. Chase the Collectors immediately after they kidnap the crew, and destroy them to save her. Leave the ship and come back to receive an email from Kelly. Enter the Captain's Quarters and call her to watch her do a dance for you.