Adding a little more to Paul's wealth of information. A lot depends on the department. Even the levels offered depend somewhat on department--which means you can do lots of fun things in your story
As far as promotion. Most decent sized departments promote up to the level of captain via a test. When a position opens up for captain admin offers all eligible lieutenants a chance to take the test. Generally, a person has to serve anywhere from 2-5 years in their current position before being eligible to take the test.
When the admin receives the test results, people are promoted out of a top percentage of results (top 10% usually). Other criteria added to test results are items in the personnel file like commendations, awards and overall perception by superiors/other officers. So technically this is supposed to assure that the most qualified overall gets promoted as it uses both a test and other criteria.
From captain on up, it's based mainly on politics and performance. You can rise pretty rapidly (although they try to be circumspect about it time-wise). As Metropolis is a big city, an inspector who achieved that rank as a detective would outrank an Inspector in charge of Records. And it'd be easier to get to inspector in the first place as a detective.
Bill Henderson is a detective and, as Paul says, would be higher profile--and he seems to be good at public relations which would get him everywhere. If you want him to move up two ranks, I'd say you could believably do it in the course of 5 years. Probably not 3 unless he single-handedly saved the president or something that huge. Departments try to be careful not to promote too quickly to keep bitterness at a minimum.
