In real life, you're usually not handed-over your job position directly; have to work it up slow, bit-by-bit. You first have to apply for a lower-level position, arrange an interview, and get into the job, somehow. If you're doing well and start developing respect among employees and bosses, you get rewarded with a promotion offer. If you accept, there you go--you're now at some leadership level. Same applies in Pokémon world, I believe. First, you register for some Gym trainer position by filling out forms, and show-off your knowledge to whoever's in-charge. When it's time for current depart, you might be able to get a hold of their position and become the new leader. After few months of gaining experience, you're then eligible to join Elite Four, and should they ever need someone new, you could apply instantly and prove if you're worthy or not.
Of course, there could always be other ways (having good relations with higher-ups, perhaps?), but right now, this is the only one that seems most logical. Well, at least to me, anyway.