Instructions
Speak to Kelly and flirt with her. Every new dialogue branch she has will give Shepard the option of being professional, cruel or flirty. Unlike the rest of the romance options, Kelly will respond to flirting from a male or a female Shepard.
When Kelly asks to have dinner with Shepard, generally after three instances of flirting, accept the offer. The dinner date will happen off screen.
Speak to Kelly again. She will offer to feed your fish for you. This advances the romance and relieves Shepard of the burden of feeding his fish every time he returns to his ship, the Normandy.
Do not engage in any other romance. Flirting with the other options will quietly end Kelly's interest in Shepard.
Save the crew. After the mission on the derelict Reaper, Shepard's crew will be kidnapped. This includes Kelly. Shepard will then be presented with the option of immediately attempting to save his crew, or continuing to prepare his strike team for the final battle. Choose to go after the crew immediately to save them. If you delay, they will die. As a side note, Legion's loyalty mission doesn't count. If you complete its loyalty mission, then immediately go to save the crew, they will still survive.
Send an escort back with Shepard's crew. During the final mission, you will find his crew alive, but injured. They will ask for one of his strike team to escort them back to the Normandy. If he does not send one, most, if not all, will die. Mordin is the best choice. He has the highest chance of getting them back to the Normandy alive and is the most likely to die permanently if Shepard takes him to the final battle.
Complete the final mission.
Go to Shepard's cabin and check his email. There will be a letter from Kelly propositioning Shepard.
Interact with the intercom on Shepard's desk to summon Kelly to his cabin. She will be wearing a tight leather outfit. Shepard may summon her whenever he wants, as he completes sides missions and DLC missions.