Yes, you can, but not easily or directly. You'd be extremely unlikely to be able to get a residence visa as a police officer. IE if they need a certain type of skill set then it's easier to get jobs and visa's to work/live there. Police officers would be unlikely to count.
If however you were a cop here for years and became a excellent detective then you may be seen as skilled enough to apply for a job or position over there. You wouldn't get a visa to go over there and train to be a cop.
If you get a great degree and can get a job in another area that they might get you a visa for, you could at a later point in your career then change and become a cop.
Going to uni there would be much simpler but in no way makes certain or even noticeably helps staying their permanently. If you did a placement year in the states, which is fully possible, on a degree here. IE an engineering degree then do the third year on placement with a US company. If you can impress that company enough they may be willing to sponsor your visa application and offer you a job in the states once you finish your degree. That is likely one of the easiest ways for a student to get a job in the states.
Another option is joining a multi national company with offices in the states. Work hard, well and apply for any positions in the US officers after a few years when they come up.
Of make loads of money, if you have loads of money and intend to spend or invest it in a business over there, then getting a visa is pretty easy.