Vincent Vega is a known gangster who just came back from Amsterdam. Mia Wallace is the wife of his boss Marsellus Wallace. Their relationship is the most zig zagged one in the history of cinema.
At first Vincent didn't really expect to find anything interesting about Mia and the initial plan was to simply spend some time with her and fake interest. However upon going out to dinner with her, he finds himself charmed by her personality sparking a friendship between the two. When he finds her overdosed on the couch, his worry was split between fear of retaliation from Marsellus as well as fear of losing his new friend. Mia in return brought out a side of him that was only seen when he is among good friends.
While not flat out stated outloud there were many hints throughout the movie that suggested they would have become more than friends. Prior to meeting her, Vincent is warned by both Jules and the Bartender in Marsellus' not to get too serious with her, implying that in other circumstances those two would have been a couple. He was willing to dance with her albeit reluctantly and during the bathroom scene prior to her overdose, he was trying to talk himself out of taking the relationship beyond platonic.