The process is rather routine, if non-trivial. They'll need to incorporate as a township, if not already incorporated. Population requirements (and others) vary state to state.
The establishment of a new police department also varies state to state, and you can read up on the various state laws governing them. Certain requirements have to be be met (mostly concerning how they are staffed). For instance, this is Ohio:
Title 7 (Municipal Corporations) §737.05
The police department of each city shall be composed of a chief of police and such other officers, patrolmen, and employees as the legislative authority thereof provides by ordinance.
The director of public safety of such city shall have the exclusive management and control of all other officers, surgeons, secretaries, clerks, and employees in the police department as provided by ordinances or resolution of such legislative authority. He may commission private policemen, who may not be in the classified list of the department, under such rules and regulations as the legislative authority prescribes.
Basically, the city government needs to pass ordinances, then hire a police chief who then has to hire out others to do the actual work. The police chief does not himself need to be a sworn law officer in some/most states (wouldn't have the power of arrest if he doesn't though).
Nothing in the statutes I'm looking at talks about having to register with the state government or anything like that either.
Such a town might find model laws/statutes with which to pass such ordinaces (basically, it's like a "fill in the blanks" bill example, so they don't have to write it from scratch).
Also of interest, in the United States, universities and transit companies (railroads, but also others) can establish their own police forces. The latter can do it according to federal law, the former having various differences state-to-state.
If you want this story in a particular location, consult the laws for that particular state. However, things tend to be similar enough that this might not be necessary