To earn the "A SMILE ON HER FACE" trophy in Detroit: Become Human, you must complete the chapter "THE PIRATE'S COVE". This is a Kara chapter, so naturally you have to make sure she survives her previous chapter "ZLATKO"; see the following trophy for guides:
From the flowchart it would appear that the chapter only has one potential ending, which is putting Alice on the carousel. Therefore just work your way through the chapter picking the options that best suit you (generally I try to keep in everyone's good books) and the trophy will unlock when you reach the flowchart screen.