you're 'allowed' one, two at most, sentence here and there. not that there's a formula for this i'm aware of, but my advice is keep it scant. also, you'll usually want to put your infodumps in the set-up/act one. that's not always possible, of course, but character-wise this shouldn't be a problem. there may be something about the character you want to reveal later that requires a little background information, though offhand i'd say that's part of your set-up, too.
a lot of people put their infodumps in a prologue. hence, you have a lot of people skipping the prologue. newbies in particular like prologues as a way to get out a lot of exposition and get on with the story without quite understanding you can show the backstory in a variety of ways.
therein lies, imo, an important trick to remember: show as much of your backstory as you can instead of telling the reader a history lesson. make no mistake, showing takes up a lot of words, but it also can lead to some interesting scenes.
by far i think the most popular method of hiding info-dumps is through dialogue. your character may come to a place or find an object that they're unfamiliar with. that's when the wizened old mentor comes in handy, to explain that to the character and thus to the reader. that's when the person under investigation finds a clue that will help her prove her innocence and must rely on her non-sexual b/f who's whiz-bang at cracking codes to 'see what he can do.' note that after these guys' practical usefulness is spent, they're often killed off.
then there's (and i hate doing this as much as i hate reading it) gathering 'round the campfire and listening to the old man tell tales. it could be dad at the dinner table or a femme fatale in the private dic's office (which would only be part of the real story, mind you). it could be a killer's confession. basically any extended chunk of dialogue that explains things away. it even extends into watching a t.v. show or corporate sales pitch on DVD. believe me, it's always obvious, but.... for what it's worth, i'd rather see the campfire version over the prologue.
if you're able to recognize your infodumps, see if you can 'show' them away first and dialogue/conversation them out second. that's my knee-jerk advice, anyway, you're welcome to disgard it. (remember, if you go the dialogue way to not do the 'well, bob, as you know...' thing, for example, 'well, bob, as you know you're married to my sister and you're working for a robotics firm.' gee, ya think bob knows who his wife is and where he works?) try to see if it's too much at once ~ like i said, you can get away with a little bit here and there.
also, if possible, combine some of the character's backstory into the infodump. then dump that into a bit of dialogue if you have to.
'i used to be a ranger in these parts. these ruins... they are cursed.'
'you were a ranger? i knew it! what happened? i thought rangers were for life.'
'they are. unless you're caught with a girl who said she was 18, but... wasn't.'
remember the line in 'star wars' where obi-wan and luke are on the bluff overlooking mos eisley? 'mos eisley spaceport. you'll not find a more wretched hive of scum and villainy. we must be cautious.' man, what a great infodump, huh? you know about the spaceport and you know that obi-wan has been there before, adding a bit to his character. plus, it's just a way cool couple of lines. in fiction form, beyond the description there's nothing more to add as evidence of what obi-wan said ('showing' how bad the place is) is forthcoming in the cantina scene.
bopping to another movie, remember the first time you watched 'raiders of the lost ark'? remember indy explaining to the g-men about the ark's history while at the college? did you understand the first blessed thing about what he was saying? i sure as hell didn't. not that it mattered, it was so entertaining to watch. in fiction, though, i probably would have tried to find a way around this scene (although ultimately it's so perfect it's hard to improve on that). i mean, could you take any bit of advice here and do that scene any better? i couldn't.
don't let me mislead you: sometimes an infodump conversation can be just the trick to use. two old women sitting around a pond, staring at the ruins of a roller coaster across the water. martha says, 'remember when the park had amusement rides?' agnes nods, 'they used to have the most gut-wrenching hot dogs.' 'yes, but if i recall, you made glenn practically spend his paycheck on them for you.' 'they made me sick, but they were the only thing even slightly edible.' 'not to mention you had a crush on the cashier. i wonder what ever happened to him?' 'he was in glenn's regiment in the war. he died storming some little hamlet in austria. i remember glenn saying so in a letter. the last letter i ever got from him. no one ever said where glenn was killed.' 'oh, yes, now i remember. poor boys.' 'you know, i never told scott about gleen.' 'you didn't? why?' 'i don't know. i think because scott couldn't fight he never had a chance to be a hero. i don't want him thinking that i had a soldier hero in my past that i thought of. it's funny, martha, i can still picture glenn's face to this day, yet i can hardly remember scott's.'
okay, that sucked. sorry for boring you. but, hey, at least it was put into a single crappy paragraph. someone may find that interesting, and it may be, i dunno, but i think it says a few things about the characters and their lives and surroundings that could have been pure infodumping and probably far less interesting (assuming this off-the-cuff example has any interesting aspects to it, lol).
flashbacks can also be used to hide infodumps. i should say 'hide' because i find a lot of flashbacks are used no better than prologues, just ways to drop in backstory for lack of a better alternative.
i'll have to think of some more. this is where specific examples that you have would come in handy.