The current (2/21/2017), general process as described on the Greenlight FAQ:
Who should submit their games to Steam Greenlight? Is there another way to submit my game to Steam?
Steam Greenlight has replaced our previous submission process. Any developer or publisher who is new to Steam and interested in submitting their game to the platform should submit their game through Steam Greenlight.
However, Valve is planning on changing the Greenlight in the near future to Steam Direct:
A better path for digital distribution
The next step in these improvements is to establish a new direct sign-up system for developers to put their games on Steam. This new path, which we’re calling “Steam Direct,” is targeted for Spring 2017 and will replace Steam Greenlight. We will ask new developers to complete a set of digital paperwork, personal or company verification, and tax documents similar to the process of applying for a bank account. Once set up, developers will pay a recoupable application fee for each new title they wish to distribute, which is intended to decrease the noise in the submission pipeline.