Die. In order to get into the Soul Society, most people must be a spirit. There are a few exceptions to this rule. Ichigo Kurosaki, "Bleach's" protagonist, has such tremendous power that he does not need to die.
Enroll in and complete Soul Reaper Academy training. The Academy teaches prospective Soul Reapers how to fight Hollows as well as the tenants of Soul Reapers.
Accept assignment to a squad. The 13 Court Guard Squads each have a captain and a lieutenant who run the squads. Within the squad, members are seated according to abilities. New members have a low seat number.
Rise in rankings. In order to rise, Soul Reapers must distinguish themselves in battle or skill level. Learning specialized attacks or proficiency with the weapon of the Soul Reapers, the Zanpakuto, will help. Soul Reapers can even go so far as to master one of the greatest attacks possible, the Bankai. While it is not a requirement for lieutenants to know how to Bankai, many lieutenants, like Renji Abarai, do.
Be appointed to the position of lieutenant by the captain of your squad. As a lieutenant, you are responsible for helping the captain maintain the squad, both in command and desk work. If a captain is unable to command -- as with Jushiro Ukitake, who suffers from tuberculosis -- you will command the unit.