To start a relationship with Vasco you will need to do his three companion quests as well as choose the correct dialogue options when prompted which is very easy to miss. Vasco is bisexual so you can romance him as either male or female.
His three companion missions are:
-A Name for a Family
-Forever a Naut
The three dialogue options that you HAVE TO CHOOSE to trigger the romance are:
"Ask him if he has any happy memories." (This is available when speaking to Vasco after making Constantine's acquaintance in New Serene.)
"Tell him that this encounter allowed him to assert himself." (This is available when speaking to Vasco after he meets his brother during Family Reunion.)
"Tell him that you wish you could sail with him again." (This is available when speaking to Vasco after Forever a Naut.)
After you've completed all three of his quests, said the correct dialogue, and have a Friendly reputation with him you can talk to him about a romance to which he asks to meet you in the bedroom of one of the de Sardet residences. Spend the night with him and the achievement will pop.