Apologies that this is so late; I’ve had spotty Internet access for most of the last week and a half.

1) Know what nonhuman “normality” is.

One common way to portray alienness is to constantly compare alien characters to human ones, of course—so the nonhuman characters have their own versions of schools, chairs, the nuclear family, unusual crops, stormy adolescence, and human art. And this works fine if your viewpoint, or your viewpoint character, is human.
The problem comes when you move into having nonhuman viewpoint characters. Why, exactly, should an immortal character describe herself as a “teenager,” or attach any particular significance to being thirteen or fifteen instead of twenty? Why should a character whose range of vision naturally extends into the infrared, ultraviolet, or both assume that the human range of vision is the standard of comparison? Why should a character from a species with three or four genders and a completely different means of reproduction assure her audience that she is exactly like a human woman and can’t wait to be a mother?
I usually see this justified with, “Well, you can’t be too alien, or your audience won’t understand anything.” But that doesn’t excuse the rest of the universe behaving as if being human were the natural or normal thing to be, when they have a substantially different reality (or, in the most extreme cases, when they’ve had no contact with humans at all and thus have no basis for the comparison).
There are ways to cope with showing the nonhuman normality—I’ll try to detail some in the rest of the rant—but first you have to make the commitment, rather than assuming that every race you demonstrate is simply a flawed version of humanity.

2) Develop rites, ceremonies, arrangements, and societies that make sense for that species.

The case this hits the hardest is with immortals. Attempting to stuff the whole of an immortal life into a simplified human paradigm, with ceremonies for birth, passage into puberty, passage into adulthood, marriage, and death, but nothing else, really doesn’t work.
What does work? Coming up with times of life that make sense for that species. Perhaps these immortals keep no track of dates at all, and an individual’s passage into adulthood happens so gradually that the only truly society-wide important events to them are birth and death (assuming they only live until someone or something kills them). Perhaps these immortals do, in fact, record the years, if only to keep track of what mortal species are doing, and thus they date themselves by external events. Perhaps there are multiple stages of life for them, following personality changes, or alterations in the body that humans don’t undergo. Perhaps individuals largely make their own choices, so that, while a marriage can be celebrated, there’s nothing to prevent the individuals involved from celebrating a parting from one another and then marrying again, and single people are not seen as inherently failures in life.
The same thing happens with species that reproduce in a different fashion. Is there a need for marriage? If not, they may still form permanent partnerships to rear young, but there is no need for the partnership to be reduced to two parents, to replicate human masculine and feminine roles perfectly, or to imitate the life of a nuclear family. (I am deeply disappointed that nuclear family patterns are seen as the way of life in so much fantasy, to the point of assuming that any child character reared with a single parent must be deprived and/or obsessed with finding his or her missing parent). Extended families of many, many different sorts could come out of a situation like this, and the more delicate and complex the means of reproduction required, the more finding suitable partners might be a focus of the society. That’s a different kind of domestic fantasy. On the other hand, maybe the reproductive patterns of the species more closely mirror those of certain large mammals, with the males and females usually living solitary lives and the mating happening only during certain seasons. Human men and women are essentially capable of having sex most of the time; members of this species might not be.
What’s the purpose of their society? It may be to protect themselves, but, on the other hand, a nonhuman society may face far different threats than humans do. Perhaps they never war with each other, but face attacks from an outside species they can’t combat; thus, they train themselves in the best means of hiding and protection rather than fighting back. Perhaps they’re a migratory species, and thus they have separate settlements in north and south, the case in LeGuin’s story “Seasons of the Ansarac.” Perhaps they’re parasites, and, when they congregate together, it’s to discuss the best way of keeping their hosts alive without killing them—and they do so only when members of their host species pass close enough together. Perhaps they’re immortal and don’t need to eat and drink, but they live by the sight of vast patterns that make the universe into an artwork, and they go about arranging other species, plants, and the earth into the patterns they need to be in to make the artwork look the prettiest.
Basically, don’t assume that the nonhumans are living poor or incomplete lives just because they have a different set of steps in life than the humans do (all those stories of bored immortals and backwards alien societies that just need human intervention seem to stem from the idea that sentience enacted on any other scale than the human one simply is not plausible).

3) Vary their cultures.

Another problem with nonhuman species can be that their cultures are far too monolithic. While humans in the same world have different nations, religions, languages, and histories, all elves live in forests, wear green robes, sing, and use bows, and all dragons are solitary and savage, with no higher ambition than hoarding treasure and fighting knights, despite their sentience. This is another effect of making the humans the center of the world. At best, nonhumans are a marginal presence in the world, or, if their society is portrayed as being superior to human society, they’re only role models to follow, without any complexities or self-contradictions to make them complicated characters. At worst, nonhumans are another kind of animal, but the kind of animal that only exists in the minds of people who don’t have much contact with them, having a few simplified behavior patterns to keep them running, and a few exaggerations for comic effect.
So show variation. If they do have different arts than human beings, that’s still no reason to have one art form consume the whole society. If they are immortal, don’t account time as important, and thus lack human “history,” what kind of plans, hopes, dreams, or plots drive their lives instead? If they have a “nature religion,” that’s still no reason they all have to be mindless slaves to a Mother Goddess. Consider how complex a “nature religion” could become that acknowledged a different deity for every species of plant and animal, or which was animistic and accepted each rock, tree, and creature as having its own sentience.
If they have elaborate systems, remember that they don’t just have to concern etiquette and court politics, the two areas that most writers think about. Think about what they may need etiquette for that humans don’t. If they can meet in the dreamworld, how do they behave there? What are the rules about entering someone else’s dreams, and how do young children learn them? If they have indestructible bodies and can harm themselves extensively, then resurrect at a moment’s notice, they may still admire people who undergo certain deaths more than others, and for reasons that human visitors wouldn’t necessarily assign at first glance.
Nonhumans may still be parts of the scenery if your story takes place primarily among the humans, but the parts of them on display should still be crafted with as much care as the human societies the book only briefly touches on.

4) Remember that their nonhuman attributes will influence multiple parts of their lives.

Perhaps every member of this species has empathy, and they can only learn slowly to damp it; in fact, children sometimes go insane from the emotional overload. How would they deal with it? I found it doubtful that they would just refer to empathy in passing now and then and otherwise act (and expect others to act) exactly like people who didn’t have it.
Social rules would accommodate emotion and the expression of it; after all, if everyone knows what you’re feeling anyway, there would be little point in attempting to keep up a stoic front. There might be some areas in which self-control is valued, but those places would automatically have to leave out children and the people poor at suppressing their empathy—and even that might not be enough when, say, two old enemies or two reunited lovers met. Conversational give-and-take would reflect it, perhaps with pauses to see what impact particularly controversial words made on the partner in the conversation. How one experienced the empathy would also need to be given consideration. If it seemed like physical sensations, such as heat for rage and the smell of vomit for extreme anxiety, people might leave the person feeling such emotions alone for a time. Lying and honesty would have totally different meanings, and those who masked their empathy or emotions might essentially deafen themselves or draw suspicious attention from others. Anyone from another species without empathy would probably find some social situations confusing, or impossible to follow. Games of chance that relied on bluffing, like poker, would become impossible, and an art form like drama that depended on acting would be complicated in interesting ways.
The same thing should happen with long life, keener senses, illusory magic, a preference for a watery environment over an airy one, wings, the ability to shapeshift, and so on. The society that refuses to make accommodations for an ability everyone has is simply stupid (and/or the writer is once again relying thoughtlessly on human norms). Again, think about what the center of normality is in your society, and then thread the abilities, problems, and neutral differences from the humans that nonhumans live with daily out from there.

5) Remember that attitudes and concerns may not match up to human ones exactly.

Another problem with trying to illustrate everything a nonhuman character says or does in exact analogies; what if this species simply doesn’t have war, or a conception of it? What if they not only know the exact time of their death but refuse to let that bother them? What if they live in harmony with fate and don’t have an obsession with individualism that assures them the highest goal is for every person to be unique and utterly free of will?
All of this is fine. And I think a skilled writer can extrapolate, again, outwards from such differences.
It may be impossible to write a story like this. But declaring it impossible beforehand is, I think, like a writer assuming that she simply can’t write protagonists of a younger age than herself when she’s never tried, or someone from a different religion. How does she actually know? She may mess up on some attempts, but others could be brilliant successes.
Since I find nonhumans the most personally fascinating part of fantasy, I may be giving this more weight than it deserves. But since I’d like to see some more attitudinal and cultural variation in fantasy anyway, I do not care.
More could be done with this, always. A story like James Tiptree Jr.’s “Love is the Plan The Plan is Death doesn’t work for everyone and has not a human character in sight, but that doesn’t mean it’s not a good story.