How do I become a bureaucrat?
First, consider whether becoming an bureaucrat is the best way for you to contribute to the project. Bureaucrat status on Commons is not an honour for being an outstanding admin, and some of the roles differ significantly - in particular the community role of "leading and guiding".
We do not have any hard and fast rules setting out the minimum period that you should have served as an administrator here, but the requirement that you be a experienced and trusted administrator means that prior to applying you should have had sufficient breadth and depth of experience here to be able to demonstrate that. If you are a bureaucrat on another project, by all means mention it, but bear in mind that off-Commons experience on its own is normally not enough.
Although not mandatory, it is a good idea to talk to existing bureaucrats to get feedback about your experience and approach. You may be able to get another bureaucrat to nominate you, although self-nominations are also allowed.
When you request bureaucratship, the community will consider your ability to fill the bureaucrat role. Individual Commons contributors may have varying expectations of candidates, and you may wish to review some archived applications to get a sense of what people are looking for. As a practical matter, users will typically want to be comfortable that you have appropriate judgement to be trusted with the bureaucrat tools, and that you have demonstrated your capacity to "lead and guide". They will also want to be satisfied that you are able to deal appropriately with sometimes sensitive information.
Bureaucrats are expected to remain reasonably active here, and non-active bureaucrats may lose the tools (including admin rights) on the same basis as any normal administrator.
The request for bureaucratship
When you are ready you can apply, or ask your nominator to apply, at Commons:Bureaucrats.