Preference: Female Inquisitors, human and elven only. Note that Cullen is an advisor in Dragon Age: Inquisition, not a companion.
Cullen only likes fully unattached suitors. No options will be presented if you are currently in a relationship, so you will have to break things off you want to have a chance with Cullen. He also doesn’t like any subject matters involving Lyrium, so sure to avoid the subject. Regular conversations and flirtations will trigger the romance.
Preference: Female Inquisitors, elven only.
Keep flirting and give Solas replies that meet his approval. Once you’ve gotten some of his Approval Rating, bring him from Skyhold to Haven and kiss him. Solas will then entrust you with a request. He would like you to get in combat with a spirit, but instead of killing it, you have to free it. This will strengthen your Approval Rating. Do not expect to win any more approvals if you fail this mission.
Continue to flirt and impress him with your replies. To finally consummate the relationship, complete the Temple of Mythal mission.