(Warning: gratuitous generalisations to follow. Because how the @#$% else are you meant to infer things from five photos unless you're willing to generalise with bold predictions based on averages and hunches about human nature? Related to the above, if you're tempted to get butthurt, you might want to reacquaint yourself with the different definitions of the words 'all', 'most', 'the average', 'the median', 'the mode', 'some', and 'a few'. Also, while these are focused on observations about women, I'm sure an equivalent list of equally rash observations could be made about men - my skepticism about human nature is quite equal opportunity)
People will pick the photos that most display what they like about themselves, given the photos they have an front of them.
This may seem entirely trivial, but consider the alternative version of what people probably should be doing, namely picking photos best calculated to appeal to the opposite sex, or even better, calculated to appeal to those specific members of the opposite sex that they'd like to attract.
Most people don't get that far. This is bad for them, but immensely useful when we'd like to understand them, because we get a very good window into their likely personality.
Let's start with some basics. This will necessarily be somewhat stream of consciousness
Who else is in most of their photos? If it's mostly just them in front of a mirror, I'm guessing that they're likely somewhat vain and narcissistic. In girls, lots of mirror selfies is often correlated with a large amount of makeup in most of said photos. What they like about themselves is their physical attractiveness. Not only that, but (related to the second part of the main thesis) it means they already have multiple photos of themselves in the mirror. It's possibly these were taken specifically for Tinder, but I wouldn't bet on it. Fairly or unfairly, I assume that people who mostly value their looks do so because they don't have a lot to offer intellectually. In other words, they're not dumb because they take mirror selfies, but it's how I'm betting nonetheless. I'm wagering they're more likely to be a princess. On the other hand, if you're after someone who's going to get really nicely made up in a cocktail dress when you go out at night, this is probably your girl.
An alternative is the person who takes most of their photos in groups of friends. This means not just that they're more sociable, but that they value that aspect of themselves. On the whole, this is not a bad thing - it tends to go with extroversion, for instance. A mix of group and individual shots is probably good. For reasons I can't articulate well, I tend to assume that someone who has primarily group photos does not have much of an interesting personality - I suspect that they think of themselves mainly in terms of their group of friends, which means that they're an example of a particular type of person, rather than being strongly themselves.
A particularly interesting twist on this, however, is when the person has only photos of themselves in groups, particularly if this includes their cover photo. This isn't a problem in terms of the fact that they hang out lots with friends, but it definitely speaks to a lack of self-awareness about something much more basic - they haven't figured out that you don't initially know who they are. Anybody that puts only group shots of themselves, particularly when they look a bit like their friends (although that's hard for most people to judge about themselves), is necessarily self-centered. They know who they are, and so it doesn't occur to them to put themselves in the position of a potential match who doesn't know who they are and is trying to figure it out. The fact that they haven't reflected on this this since coming across the same problem with members of the opposite sex also speaks to low self-awareness.
As a counterpoint to this, pay attention to people whose photos are a grainy picture of their face. This means that a) the photos of themselves where they think they look the best are those in groups, and b) they're self-aware enough to not put the photo of the whole group. But more importantly, it means they have taken very few photos of themselves outside of group situations where someone brought out a camera. This suggests they're likely to be low maintenance and probably not very sentimental. I think this is actually not a bad signal, at least in my preference ordering. But it also signals that they aren't committed enough to dating or the app to take better photos of themselves.
The opposite, of course, is someone who has good somewhat artsy photos of themselves.I like this in moderation. It signals creativity and a sense of them liking something artistic about themselves. It also generally shows some degree of forethought. I would also wager that it signals a non-trivial degree of confidence, because truly good arthouse photos of you are hard to take by yourself. As a consequence, they probably had to be confident enough to have someone else stick a camera up close to their face, and displayed enough forward planning to ask the person to do this for them. Plus if they have any kind of photographic flair (such as a sharp focus from a low f-stop lens) then they have a digital SLR, which probably speaks to being at least middle class. The ones I like are where the rest of the artistic detail in the photo is done nicely, or even if one of the photos is mainly a nature shot. On the other hand, anything that looks explicitly like instagram modifications (particularly when applied to photos that mostly feature their face prominently) or other related things suggest that they also enjoy attention, and that these were taken mostly for the internet bubbas.
Another key metric is how many of their photos they're smiling in. Again, think self-perception. Most people prefer to think of themselves as happy, so will generally pick photos where they're smiling. For me, I place a surprisingly large weight on someone who has a big genuine smile in their photos. Happy wife, happy life, as they say. Someone who goes mainly with serious-looking pouty faces is probably deliberately aping the model photos they've seen, if they look like posed faces. This for me is a minus, but again, your mileage may vary. Other people tend to look serious, which I assume to mean that they're just not very much fun. I mean, if you can't even think of yourself as fun, it's probably going to be pretty hard for the rest of the world. Someone who has mainly photos of themselves pulling funny faces is probably self-conscious, and the stupid faces are a defense mechanism against the fact that they're uncomfortable with most of the photos of themselves. Someone who takes too many photos of themselves laughing seems oddly to me a vague warning sign - they like to be jovial, but I have a sense that they also expect the world (and you) to entertain them, which suggests the possibility of them being a bit entitled.
You also learn something from what they're doing in their photos. Someone who has mostly photos of themselves going out will be different from someone who has mostly photos of themselves going snorkeling, or who has photos of themselves in front of famous monuments around the world. The latter two are likely to be particularly noteworthy, because they're almost certainly not taken because they're the most flattering photo of the person's physical features. In other words, if you take a zoomed out photo of yourself rock-climbing, this only makes sense in order to convey the message that you like being active. Night-life photos could go other way - it could be that you like partying a lot, or just that you think you look good in a short dress (although the two tend to be correlated anyway).
Finally, there's the obvious separating equilibrium that anyone who doesn't display a full-body photo is probably overweight, but you didn't need me to tell you that one.
This is just a sample of the correlation-fu I'd be busting out for a mail-order bride, so from this you can extrapolate to guessing what the full-retard looks like. There's also a whole separate post to be written about what someone's bio says about them, but I feel this is enough to get you started.
For the time being, I leave as an exercise for the reader (in the comments if you're so minded) the task of forming similar personality estimates based on:
b) which things may be markers of socioeconomic status
c) the demographic diversity of their friends, related to both b) and likely political opinions,
d) their attractiveness relative to their friends, related to insecurity and self-awareness.
e) the gender ratio in their photos.
and as the bonus round
f) what to infer if they have a photo of themselves kissing their dog.