May we investigate or ask people about a prospective spouse's background?

Yes, and it is encouraged, because marriage is a major decision that requires accurate information.
Ask people who know them: the local imam, employer, colleagues, neighbours or friends, about their prayer, character, financial responsibility and how they treat their family. The Prophet ﷺ himself gave Fatimah bint Qais honest information about the two men who had proposed to her. Do it with proper manners: intend to make a decision, not to dig up faults; do not pry into private matters unlawfully (such as hacking a phone); and keep what you learn confidential. Public records can also be checked, for example marital status at the religious affairs office when applying to marry. See criteria.