Background Goals are instances where you or your party does something that aligns with their background - an early example of this is making sure nobody dies at the battle outside the Grove, which will earn you one inspiration point for Lae'zel who has a Soldier background.
You have 2 options for this:
1) Check the official BG3 wiki for your background (if you don't mind spoilers)
2) Play as the Dark Urge and embrace it; i.e. kill everyone and everything. It's the background with the most obvious background goals - they usually have unique dialogue options, even if they don't actually say [Dark Urge]