If someone wants to become Stack Overflow staff member how can they do so?
You get hired - you can find the listings here for currently open jobs, apply, meet the criteria listed, do interviews and such.
In general, all other things considered, I'm told that a good cover letter is helpful, alongside a relevant resume. If someone in staff feels you're a good fit, you might also be able to get them to refer you.
What are the criteria for choosing a staff member?
I've never gotten as far as 'serious' interviews. Apparently good cover letters help. Traditionally, knowing the platform well has been a key criteria and might be helpful. Many roles might not involve actually working with the community, so it depends
I suspect it would very much depend on the role and team, since SE does have a significant number of staff in sales, admin and other non public Q&A roles.
You can find a baseline expectation for those specific roles in the respective listings.
Just like for mods, is there any reputation threshold or badges you need to have in order to become a staff member?
Generally no. We've historically had folks with not much reputation and badges come in and be spectacular members of staff. While I personally feel we need to hire 'in community' for some roles - it should never be a defining criteria. We've even at one point had an entire sub team of community managers who were from outside the community and helped build up many of the sites that were started at the time.