The word "doppelganger", which means something like "double-walker" in German, only dates back to the late 18th century. However, the general concept of a spooky double of a person is a common one in mythology, folk tales, and just plain ghost stories going back for thousands of years. It's also a common theme in more recent fiction, with perhaps the alien in the 1938 novella Who Goes There? being a particular inspiration for the D&D version. (The story was later remade as a film titled The Thing, although the black-and-white version Gygax would have been familiar with at the time features quite a different sort of monster).
It's hard to know what we're supposed to be looking at in the 1E illustration of the doppelganger. Is this the "true form" of the creature? Is it, perhaps, partway through transforming from one shape to another? Or adopting a freakish form just to be startling to someone? Is that a literal glow around its head, or an artistic convention representing its psychic powers? Whatever it is, it looks nothing like the versions seen in later editions.
We can, however, learn a few things from the stat block and accompanying text. Doppelgangers are significantly tougher than the average human, either because of innate combat skill or some innate resistance to injury. For instance, with their mutable form, it's possible that they can seal injuries up - not enough to regenerate, but enough to minimise blood loss and so make a wound less serious. Or perhaps they lack, or can at least adjust the position of, internal organs, making them harder to strike a lethal blow on. Their high damage rating (unrelated to weapon use in this edition) also implies that they are much stronger than humans.
They live and travel in surprisingly large groups, which could perhaps represent extended families, although there's no indication of how their society works. Indeed, almost the only thing we know about them is that, despite not being "evil" they regularly kill humans and take their place. From a game-mechanical perspective, that makes their purpose clear enough; they are monsters that work through stealth and confusion rather than brute force. But what they gain from doing this in-world is less apparent.
It's in 2E that we're told for the first time what a doppelganger looks like in its natural form. They are specifically stated to be hairless grey humanoids with a tough hide. The illustration shows elongated digits and ears, dead white eyes, and a generally cadaverous form, although it's possible that it's just starting to shapeshift into, say, an elf.
They are also given a clearer motivation than before, seeking to imitate wealthy people so that they can live in luxury, effectively as parasites on society. On the other hand, while they are still not described as evil, we are told that they regularly work for evil people (and not, apparently, good ones) as spies. There's also some hint of a cultural split between comfort-loving, fundamentally lazy, urban doppelgangers and more aggressive groups living as bandits in the wilderness.
We're told that they're "tribal", and even that they all belong to one single tribe, although quite what this means in the context of a race so scattered and widespread is unclear.
One key innovation in the 3E version is the change of spelling from the non-standard "doppleganger" of the first two editions to the usual English variant of the German Doppelgänger. The look also changes slightly, for whatever that may mean for a creature that's inherently mutable, with a larger head, green or yellow eyes with horizontal pupils, and no visible ears, pointed or otherwise.
This version, while still stronger than a typical human, seems to be far less so than those in earlier editions, judging from the damage it delivers with its fists. They are, as might be expected, agile and highly adept at deception although they don't seem to be as stealthy as they were before. They also gain the ability to see in the dark for the first time, and we learn that they don't have a native language, typically speaking the local human one instead.
Although we are explicitly reminded that they aren't evil, we're told that they regard other people as mere playthings to be manipulated solely for their own selfish ends. Which, and maybe this is just me, does sound a bit evil from where I'm standing.
The Doppelganger shown here is more muscular than previous versions, and, unlike the 3E one, has ears and no apparent pupils - or mouth. Up until 3E, they were always immune to sleep-inducing magics, but that particular benefit is lost in this version. In physical terms, they are more agile than in earlier editions, but they are no longer particularly strong and don't have the slightly enhanced intelligence of 3E.
The large groups of the first two editions have gone, and even the small groups of three or four described in 3E now appear to be unusual, with doppelgangers often working alone. Most other changes reflect the ruleset, emphasising their existing abilities to strike from ambush and maximise the element of surprise.
The fact that the doppelgangers of the first three editions are immune to effects that would cause other beings to fall asleep might be taken to mean that, like elves, they do not naturally sleep themselves. However, there is nothing else to indicate this, and their classification as "monstrous humanoids" in 3E would, absent a specific statement to the contrary, imply that they do.
More likely, then, it's a side-effect of their telepathic powers, which are presumably also responsible for their immunity to mind-control magics, which is retained in 5E. (Notably, 4E doppelgangers have no psychic powers and also lack both immunities). The exact details of those powers don't really change between editions, except to line up with other rules on spells and, as described, are insufficient to allow the doppelganger to fully imitate a person whose mind they have read, although they surely help to some extent.
But it's obviously the signature shape-shifting power of doppelgangers that is the most significant thing about them. It's clear that doppelgangers can imitate humans, dwarves, elves, orcs and so on, but the first two editions indicate that they cannot imitate people less than four feet tall. That would include halflings and human children, but this limitation disappears in 3E.
Instead, the requirement given there, and in subsequent editions, is that the subject must be 'humanoid'; there is a size limit, but no standard humanoid races exceed it anyway. This includes non-mammalian races such as lizardmen, while excluding giants, sprites and so on. More surprisingly, a literal reading of the rules implies that doppelgangers can imitate mermen but not some actual bipedal beings such as wights or mind flayers. Not that they'd want to do either, one suspects.
We also know that the shapeshifting isn't some kind of illusion, but an actual physical change. But how it works seems to vary radically between the first two editions and the later ones. From 3E onwards, the implication is that the doppelganger is, at some level, a regular humanoid being and presumably has most of the usual internal organs. The illustrations seem to confirm, for instance, a skeletal structure and musculature broadly similar to those of humans, elves, and the like.
But it isn't just the skin tone and texture that changes when they shapeshift. The skeleton itself must also be mutable, in order to imitate different facial features, never mind in order to alter things like the length of arms and legs. Furthermore, they can sprout hair that they don't normally possess, reshape their teeth into tusks when imitating orcs, and so on.
But can they change their weight? 3E states that, in their natural form, a doppelganger weighs about the same as a human, but is that still true if they are imitating, say, a halfling? If the effect were due to something like a polymorph spell there wouldn't be an issue, but it can't be magically dispelled, so I'd argue that it probably isn't. If so, to imitate a halfling, a doppelganger must be able to compactify its flesh, maintaining the same mass - in most circumstances, this wouldn't be noticeable unless somebody tried to lift them, but the increased density might be an issue while swimming.
In the first two editions, however, it's a good deal stranger. That's because, in those editions, doppelgangers can also imitate the clothing and even carried equipment of those they duplicate. Being able to change your skin so that it looks like a shirt of chain armour is one thing, but equipment? You might not notice that somebody never undresses when you're around, but if they never put down anything else they're carrying, because it's physically part of their body, it's hard to see how the deception can work for long. And if they can... well, that's weird, to put it mildly.
It's not surprising that this idea was later ditched.
In addition to fully-formed facial features, there is another feature that doppelgangers in 3E and 5E clearly lack: genitals. While they can presumably form them at will, this does highlight the question of how the race reproduces.
In the world of Eberron, originally written for 3E, doppelgangers seemingly reproduce in the usual mammalian fashion. We know this because they can also crossbreed with humans, creating the race of changelings. By implication, therefore, doppelgangers must normally reproduce by breeding with each other, although it's entirely possible that they are hermaphrodites.
The 5E Monster Manual, however, gives a different take on its non-Eberron doppelgangers. Here, the race can only perpetuate itself by breeding with humanoids (presumably any of them, including, say, dragonborn). They are said to only ever transform into males to do this, although this may not be a biological limitation as much as being too lazy to want to bother with pregnancy and child-rearing. On the other hand, they obviously can imitate individuals of either sex and, being hedonistic, it seems likely that this is something they would take advantage of from time to time.
Speaking of which, their imitation power seems to be based on observation so, unless they've been watching the target carefully, if they want to imitate a specific person, they are likely to omit things such as hidden birthmarks they aren't aware of... or indeed, make other anatomical errors that an intimate partner might notice. Much easier, as the text implies, to take on the form of a stranger.
Aside from the actual shapeshifting powers, and any ability to read surface thoughts, doppelgangers are typically seen in D&D as more agile than regular humans, but otherwise physically similar. (In some editions, as noted, they might also be slightly stronger). Other than this, in most other systems, the key is to emphasising their powers of deception and persuasion.
If there's a stat that's equivalent to Charisma, it's likely above the human average, but, for the most part, this seems to be based on skill and experience. This means that, in systems that have specific skills for this sort of thing, doppelgangers should be good at Fast Talk, Convince, or whatever the relevant equivalent is, as well as Acting and any observation skills that might make it easier for them to pull off the con.
In most non-D&D based systems, however, the question of whether their alignment is really Neutral, or something worse, is one that won't arise.