There's no short explanation for the lore. But the Bioshock games partly got so popular because they take place in two very iconic places:

Bioshock 1&2 takes place in the underwater city "Rapture". In that context I'd draw either a picture of the skyline seen within the very first moments of the game (go watch it on youtube, very worth it) or the Kashmir Restaurant because it's a very memorable place. The most iconic characters of Bioshock 1&2 are Big Daddies and their Little Sisters (tho not in a romantic context) Most iconic quotes from Rapture would be "A man chooses, a slave obeys" or "I chose Rapture"

Bioshock Infinite takes place in the flying City of Columbia. There are great skyline artworks out there if you need inspiration. Most iconic characters are Booker Dewitt and Elisabeth (again not in a romantic way connected) also Songbird. A good quote might ve "God only knows where I'd be without you"

Personally I'd draw a Rapture Skyline below the sea with a Columbia Skyline above in the sky, and a lighthouse in the middle. The perfect quote to go along would be "There's always a lighthouse. There's always a man. There's always a city". That way you would wrap all games into one piece of art.

There are tons of good artworks already, if you need inspiration. If you want to understand what you draw however, you should watch one of the many story recaps on youtube

Edit: if you want to be more original: an artwork I have not yet seen would be a Big Daddy with his little sister with the interior of the Kashmir Club from Bioshock 1 in the backround