According to the comics (as reference by Wikipedia) Mystique's abilities are psionic.
A photograph will give her enough to look like the person. She's trained and practiced enough to mimic people well - she can adopt their common behaviors and verbal/physical tics quite easily, given reference.
Once Mystique is in areas where her subject has been in the past, it's likely that she picks up more details on them. To a trained observer, the way one's personal areas are laid out and organized (or not) can give insight into their mental state. It's guesswork, but it's educated guesswork.
The fact that her abilities are psionic probably helps - 'psionic' is a catch-all term that Marvel tends to use for 'freaky things that shouldn't work, but do, but aren't magic or technology'.
We have evidence (in comics and on-screen) that Mystique CAN duplicate a person well enough to copy their fingerprints and retinal scans. 'How' is a difficult question, but I expect the likely explanation you would get from a writer (as there is no canon answer I can find) is that she either absorbs their DNA from physical contact (either with them or shed skin/hair in the vicinity) or better mimics them by absorbing their psionic imprint on the areas they frequent.
Neither of these really make sense, but then again Mystique's mimicry is pretty nonsensical in and of itself.