"Legal" in D&D is "whatever the DM of your game decides to allow". Talk to your DM.
Don't confuse "backstory" with "background".
Choose a
background from the Player's Handbook. The whole of Chapter 4 is about backgrounds. You need to choose one of those. If you don't know what to choose, take the Charlatan background on p.128, which is the recommended one for Rogues.
The backgrounds in the PHB are generic and will fit into almost any game of D&D. The Slayer background you linked to is not in the PHB and may or may not make sense in the context of your particular game. Dedicating your character to slaying fiends, for example, is useless if there are no fiends in your DM's game. That's why it says "work with your DM".
Talk to your DM. The one who is going to be running the game that you are making your character
for. He will already have decided what the "world" is like that you are going to be adventuring in. Then create a
backstory connecting your character to that world. You can't do that in a vacuum; you can't do it if you don't know anything about that world. Also, some DM's like to see a long, detailed backstory because it gives them ideas for developing their story, whereas others don't want it and will ignore it or ask you for a simple, one-line backstory because they are running a module where the story is already largely worked out.
- Talk to your DM.
- Talk to your DM.
- Talk to your DM.