A fellow author of mine posts a regular video diary where he talks about various things writerly, and in a recent post he described how one of the ways to introduce a character to the reader in a galvanizing way is to make the character the recipient of undeserved misfortune. Emphasis on the undeserved; you want to make it clear they didn't have this coming, and so we feel for them and want to see them do well. Harry Potter is one such example; the kid's living under the stairs when he first gets his invite to Hogwarts.
A caveat I'd put on all this: there's a few pitfalls in making a character's undeserved misfortune into the way we meet them. One is in having the flavor of their misfortune be the only interesting thing about them. A person is not interesting because they suffer, but how they respond to suffering, positively or negatively. (Macon Leary of The Accidental Tourist responds to the death of his son not just by becoming emotionally muffed, but by having the emotional numbness he's always had in some form, but at least managed well, re-emerge. How he deals with that becomes the substance of the story.)
Everybody suffers to some degree, and so to single someone out because suffering has visited them is not by itself why we pay attention them. We are interested in their suffering as we would a doctor tending to a patient's broken leg, but the broken leg is not the most important part of the person.
The other, more obvious one, is to resist the temptation to throw suffering at someone because it "builds character" or "gives them something to do", etc. Most writers know the stock formula for action in a story: put a character up a tree and throw rocks at them. (I thought immediately of Martin Scorsese's After Hours, where a hapless Manhattanite becomes trapped in a black-comedy downtown landscape of weirdoes and mounting danger.) The formula, though, puts too much emphasis on simply attacking the character, when the point of all that attack is to get them to show what they're made of — to express character through action. I feel the way the lesson is imparted puts too much emphasis on the throwing of rocks, and not enough on what they themselves bring to the fight.
We take it or granted in a story that if a story is being told about someone, it's because something is happening in their life that is worthy of comment, and that something is typically them being in a bad way for whatever reason. The danger is to have the person's suffering become the mode of distinction for the story; go too far down that road and you're left with one-upsmanship.
Some of the criticism surrounding Hanya Yanagihara's A Little Life revolves around how one of the characters is forced, over the course of his life, to run a gauntlet of suffering that beggars plausibility. It's the sort of thing that authors often defend with some variation of the formula "these things happen in real life, too", but that misses the point. Many things "in real life" do not work on the page anyway. If the point is to echo the suffering of someone who once lived under staggeringly awful circumstances, then again, it might be best not to make their suffering into a story device; you'd be talking about the part of them they would probably find least interesting themselves.