I need to write something from the perspective of a character who sees colours differently from humans, and in particular can see multiple shades of ultraviolet. This isn't unique, their entire species sees like this. Can anyone suggest language or writing techniques to make that work?
This seems like the kind of thing you don't dwell on, but make a few pointed remarks on; and perhaps use as tool to get around certain obstacles.
Here are my suggestions in order:
Figure out what the world would actually look like if you could see ultra violet. If he's a POV character, come up with some terms that kind of make sense. One science fiction book I read called one of those out of spectrum colors "plaid". You can't exactly do that since you'd be infringing on that work, but it was a good idea because it gave the readers something they could attach themselves to that was distinct from known colors.
Once you know what that looks like, you can determine how it would shape culture. Something that fundamental will influence the way your aliens act, but it's unlikely to be an overt fixation since it's "normal". Maybe the overly religious would make something out of it, but your normal business types probably wouldn't.
Typically, the best way to approach writing the other is to show how similar they are so that the differences feel alien, but your reader can still connect. This alien can likely still walk, eat, sleep and all manner of other things we do. It's probably best to just have a relatable alien who happens to be able to see into another spectrum and sometimes that's cool or gets mentioned.
It's good you are thinking about this, but in practice it turns out that writing these sorts of things is more about making astute observations and showing a slight difference in the way your alien thinks, behaves or perceives the world than it is a matter of language/writing technique. IE, a lighter touch tends to be stronger than anything else.
My next recommend would be that you read A Fire Upon the Deep by Vernor Vinge if you have not. It's a good read with strong examples of showing something even more alien than you are here. In the following book he makes spiders seem downright cuddly. Both are good reads for examples of what you're trying to do, and he won a Hugo; so he knows what he's doing.
Kirk answer is very good however, I would like to add the following points:
Most of humans can see "three" colors. But some have color blindness, so they see less. You can use their testimony to see what change when you have color blindness and when you haven't.
Recently, we discovered that some people see don't see "three" colors but "four". It's call being tetrachromat. So, what you are trying to write about exist and you can inspire your work for they expirence.
Reading testimony might help getting your work more realistic.
Kirk makes a good point in his answer, but I feel it is a little hidden so I would like to expand on it a bit.
A species who can see ultraviolet wavelengths, would likely have a different vocabulary for this.
Think about our color names: we have invented completely different words for the shades of the electromagnetic (EM) spectrum we can see with our good old biological eyes: "red", "blue", "purple", "brown", "yellow" -- all those correspond to very narrow bands of wavelengths. By contrast, we basically group everything beyond that into two groups: below (infra)-red and above (ultra)-violet*.
It would make sense that species who can see some frequencies in the ultra-violet and have developed a language to describe colors would have assigned one or probably several different names to it - for example the "plaid" that Kirk mentions. Likely they would use these names in much the same way as we do: "that skirt is a beautiful dark-plaid" or "that's not plaid, that's purple!".
Since I am not really a writer -- consider this answer more a contribution from a world-building perspective -- I will limit my writing advice to this:
Use these colors as naturally as possible -- when writing a book from the human perspective you would not say "the flower was of a beautiful color that dogs cannot distinguish" but "the flower was red".
If you need to bring up the fact that some of these colors are ultraviolet you can either have an outside observer comment on it (e.g. how he cannot understand why they use two different names for the same shade of black), or if your world allows you can introduce a scientific viewpoint (e.g. a character commenting "oh yeah, your eyes have of course not evolved to see that wavelength).
Finally, note that you can also reverse the situation: perhaps there is fewer need to see red colors for this character, and in turn they can be surprised why humans are so enchanted by these boring roses.
*) Technically speaking there are some classes there as well, such as radio waves, micro-waves, X-rays, etc. - but they are mostly based on their application and much broader - literally - than "red".
You can make up a completely alien analogy to viewing colors. Like describing the colors by their frequency using musical notation.
There are studies in chromophonics that correlate sounds with colors.
One such scholar is professor and orchestral conductor Jorge Antunes. (disclosure: I studied under him).
Here is Another work that correlates music and sounds.
This is only an example of replacing one kind of notation by another.
It sounds like there is the potential for contrasting the experience of the species that can see UV and humans (or another species that sees a different spectrum).
We have a real-world example: some species of birds can see UV wavelengths. Two birds may look identical to a human, but are thought to look different when UV wavelengths are considered.
UVS vision can be useful for courtship. Birds that do not exhibit sexual dichromatism in visible wavelengths are sometimes distinguished by the presence of ultraviolet reflective patches on their feathers.
As another answer mentions, I wouldn't dwell on this as a major plot point. Still, it has world-building merit by adding "flavor" to the species, and could be the source of minor difficulties (or even amusing incidents) for a character.
"Different" vision doesn't have to only pertain to visible wavelengths. The magnetic compass orientation of birds may also be light-dependent.
Consider looking at how we do colors in real life. Not all cultures "see" the same colors we do. For example, some cultures do not have a different word for blue and green. Why? Well, it's always hard to say, but the general argument is that they simply didn't need it. It wasn't useful to them for specifying objects in life. They didn't find the need to disambiguate what we would call a blue thing or a green thing.
A fascinating side effect of this is that this language affects our perception. Studies were done with cultures with different color words involving grouping color tiles that "look the same." Reliably cultures which did not have distinct words for two colors tended to have fewer different piles, suggesting that they truly perceived fewer colors in that region.
Accordingly, the best approach I can think of is to look at what value is had from distinguishing different spectra, and label colors from there.
One of the strongest advantages of the medium of the written word (as opposed to, say, movies) is that you can describe something that is impossible to experience, and readers will still accept it.
Terry Pratchett's novel The Color of Magic has several great examples of describing a color beyond what we'd be able to see. Notice how all of his descriptions focus on the subjective experience of seeing the color and its importance within the setting, even though you couldn't picture what octarine looks like in your mind's eye:
It was octarine, the colour of magic. It was alive and glowing and vibrant and it was the undisputed pigment of the imagination, because wherever it appeared it was a sign that mere matter was a servant of the powers of the magical mind. It was enchantment itself.
But Rincewind always thought it looked a sort of greenish-purple.
[W]izards, even failed wizards, have in addition to rods and cones in their eyeballs the tiny octagons that enable them to see into the far octarine, the basic colour of which all other colours are merely pale shadows impinging on normal four-dimensional space. It is said to be a sort of fluorescent greenish-yellow purple.
This whole question is the basis to any discussion of Qualia. The idea is basically that, pick a color, any color, and describe it visually without using items that are that color. For example, what does green look like? Well, we tend to say it is the color of grass, or the color of a pear, or the skin of a watermelon... but these do not tell us what green looks like, but rather what is universally accepted as green. How do we know that green is always the same color when we discuss the matter? Could my green be your purple? We have no way of universally describing color beyond items that we universally agree are that color.
With this in mind, an alien seeing the color of Ultraviolet might have a word to describe the color, let's call it Quave, but it might be next to impossible to actually communicate that the color Quave isn't the color that humans are seeing without saying Quave exists in one object and not the other. Say for example, the Alien points out that the use of Quave in a piece of alien art is one of the reasons it's considered their Mona Lisa... where as the Humans see a blank paper that looks white to their eyes.
Conversely, the Aliens would have no concept of Red... and would not see the Red Stripes on the U.S. Flag and think Americans mad for saying that there are stripes at all.
One of the best uses of describing color that is percieved differently is the book "The Giver" which creates a society that has been bred to see entirely in gray-scale. Objects have various shades of black and white. The protagonist is realized as special by the titular character because the Protagonist has an odd fascination with an apple, which is described as having something odd happen to it that no one else sees... turns out the Giver and his successor (The Receiver) still retain the ability to see in color and that the strange thing that happens to the apple is he sees it as red for the first time... and until more color is added, the protagonist has faint glimpse of red in human faces... which the Giver explains that there is pink, which only looks red because he has not begun seeing the other colors blended into a human face.