In short, to increase your notoriety, you must commit crimes, and you must do it at a young age. This age will be age eight, as this is when the first set of crimes will unlock. When this happens, you will want to commit the following crimes as much as possible without getting caught each year.
- Porch Pirate
Keep it up until you reach the age of sixteen. After this point, more violent crimes will open up, and you will want to switch to doing them in conjunction with the ones listed above.
Related: How to join a gang in Bitlife
As more crimes unlock, keep doing them every year, and you will build up notoriety. This will also allow you to eventually join the mafia, and when you do, all you need to do to increase your notoriety is the same thing, just now for the mafia.
That being said, commit a ton of crimes and do various actions for your crime family, and you will increase your notoriety in the mafia.