Kelly is something of a secret romance option in Mass Effect 2. Unlike other romance options, there are no loyalty missions for players to complete, and no real benchmark moment in the development of the relationship. Kelly can be romanced no matter the gender of the player character, and you can romance Kelly without interfering with other romance options.
The type of relationship you have with Kelly is actually set from the very first conversation you have with her. When speaking with Kelly for the first time, select the “Are you happy here?” option, and then “We won’t”. When Kelly talks about closing her eyes and letting Shepherd catch her, choose the “I’d embrace you” to set the relationship onto the romance path.
You will eventually be able to invite Kelly to dinner, and her Kelly will offer to look after your fish. To ensure this happens, you need to always be positive with Kelly in every interaction. Eventually, Kelly will express that she wants to spend more time with you, and you can select the “Let’s have dinner” option. After that, you will need to get Kelly through the suicide mission. After the mission, speak with her and select “Are you okay?”
The next time you leave and return to the Normandy, you will have a private message from Kelly at your terminal, and you can invite her to your cabin. After the cabin scene, the romance is effectively concluded.