How would the police question someone in 1890?

It really depends, were they a suspect in a crime? Are they being held in jail? Are they just a witness? Back then, it would also depend on where they were, things were done differently in say New York City vs Tucson, Arizona. Although not exactly legal back then people suspected of crimes were often beaten and poorly treated during interrogation, the courts would either ignore such tactics or declare them standard acceptable police tactics (the laws, rules and regulations we have today weren't in place in 1890). Forensics evidence was basically non existent and as a result many innocent people were convicted and many guilty were never charged or walked away scot free. Back then it would have never occurred to an officer to take a suspects clothing for testing (since they couldn't test for anything anyways) or examine said suspects body for cuts, scratches, other wounds etc... as is usually done today. If a suspect had blood on his shirt, they were guilty, there was no way to tell if came from a nose bleed, animal, accidental cut or the victim, it was just blood on a shirt therefore he must be guilty. How they questioned witnesses isn't really all that different than today, "what did you see, when did you see it, could you recognize the person" etc... However, whether in a big city or small town or the "wild west" witnesses could easily be bought off, pressured into, or told / easily convinced that they saw something / someone that they didn't. So for your story, (assuming your character is a suspect in a crime) I would say the experience wouldn't be pleasant at times, although nice officers did exist then (as they do now), and one could beat and humiliate him while the other tries to reason with him. Most of the questions would revolve things like eye witness reports and say the suspect had a past feud or fight with the victim. I would recommend researching people like Wyatt Earp and Doc Holliday for more info and insight, and even though he was a few decades later Elliot Ness. Their stories could tell you how law enforcement used to operate in the 1890's. Hoped this helped.

The Encyclopaedia Britannica link below has a very good article about the formation of the police force both in the US and UK, which should help you.

Sorry, you are too late, well over 100 years too late. Nobody is around that old in order to question them about anything.