If you want to join the Italian Mafia in Bitlife, you will need to build up notoriety by committing crimes from a young age all the way until you are 18. Once you reach 18, you should be able then to join the mafia via the special careers tab.
Under this tab, you will then need to locate the criminal organization that says either Sicilian mob or mafia and apply to it. However, keep in mind if the notoriety bar is high, you may not be able to join and will have to pick a different mob with a lower bar or try to join more than once. If it doesn't work, you will need to commit more crimes successfully or serve at least one prison sentence for committing a crime.
That said, do this, and you should be able to join the Sicilian mafia, and when you do, you can get promoted by committing crimes for your family. The best and fastest way to earn respect and get promoted in the mafia would be to commit murders for the organization and other violent crimes. Keep this up, and you should make underboss or even boss after a few years.
However, be careful as there is a high chance you might get caught committing crimes. In the event of this, you can always try restarting or closing the app to prevent the event from saving.
But if you choose to get caught, you can always become a rat to get out of going to jail. If you become a rat, you will then need to collect evidence against your former colleagues in the mob. You can do this by click on their names under the mafia member's list in jobs and finding an option called collect evidence.
Once you find the option to collect evidence, you will need to keep doing it until a bar is filled - when the bar is filled, you will be notified that your evidence has lead to convictions.
But that's to say if you get that far, as you may get accused to being a rat, which you will need to kill the accuser before he tells other members of your mob, and you get killed yourself. Avoid this fate at all costs, and you will be offered a chance to join witness protection when it is all over with.
For more on Bitlife, check out both How to increase your notoriety in the Mafia in BitLife and How to join the Mob in Bitlife – Bitlife Mafia guide