Liam Costa is the first companion that will join your party, as well as the first character you can romance with. However, he, as a romance option, is only available to the female character, Sara Ryder - while playing as Scott you won't be able to flirt with him and the relationship will be limited to friendship. Liam Costa is a character you can have close relations with - by engaging in that relationship you won't be able to flirt and romance with other characters.
Note - before engaging in a close relationship you can save the game. Afterwards, you can go through the rest of the romance, load the saved game and engage into a relationship with a completely different character. The option to romance other characters is locked only when you are engaged in a close relationship with somebody.
Below you can find a table presenting all of Liam's loyalty missions, coupled with the requirements that must be met to unlock them. Detailed description of every single one of them can be found in "Allies and relationships" section of this guide.
How to unlock
Liam Costa: Outpost Worlds
Liam Costa: Armor Diplomacy
Reach Aya and get back to Tempest. Afterwards, speak with Liam to start the quest.
Liam Costa: Day Out on Aya
Complete "A Trail of Hope" mission and visit Vault on Aya.
Liam Costa: All In
Complete "Liam Costa: Armor Diplomacy" mission.
Liam Costa: Community
Complete "Liam Costa: All In" mission.
Liam Costa: Something for Us
You have to play as Sara Ryder and engage in a close relationship with Liam during "Liam Costa: Community" mission.
Note: the option to engage in a close relationship will appear during Liam Costa: Community loyalty mission. Remember that this choice will prevent you from flirting and romancing other characters. If you still want to romance with others and at the same time see the final parts the romance with Liam, save the game before starting "Liam Costa: Community" mission and load it later if needed.