If you want to become an underboss in the 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 to then 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.
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That said, do this, and you should be able to join a 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 by commiting crimes for the mafia would be to commit murders and other violent crimes.
It would also help to keep doing these crimes as much as possible without getting caught, as it will make you get promoted to underboss in 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.
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