Writting a piece of music using the chromatic scale would be interesting, but I can't think of a way to start it? What kind of structure would a piece have? Is it all subjective and up to the composer or are there similarities between them?
Also, would it have a tonic, supertonic, mediant, etc?
I apologize ahead of time if these questions are too subjective.
Twelve-Tone Tonality by George Perle may be useful.
As to 'what kind of structure a piece would have', remember, you can structure music not only with pitches, but also time. I deem time to be an even more basic consideration than pitch in music. So ask: is there a regular beat? how many parts to divide a beat into? can you use two kinds of durations to form a pattern regularly for counting larger units of time? How many of each for each count? how about forming such count patterns with three or more kinds of durations? do you use a regular meter? how often do you change meter? how many units of meter does each phrase last? do you vary your phrases in length? can you use different tempos in different parts of each phrase in a regular way? can you do it in a non-regular way? how many phrases are there in each section? do you group phrases into sections with tempo? how many sections can you have? how does the different usages of (subdivisions and patterns of) beats, meter and phrases characterize your sections?
Most importantly, how do they feel to you? That shapes 'how you compose'.
What (physical, therefore sonic) resources do you use? This is what makes your writing more than something like solving a crossword puzzle. Do you plan do use all of your resources all the time? That would be boring, if not tiring, wouldn't it? When Schoenberg intentionally used all his pitch-class resources regularly, to avoid your boredom, he structured his time and used his instruments in a lot of different patterns and combinations (I said pitch-class instead of pitch there because he also played with how to use the same class of pitch in different octaves) WITH A LOT OF FREEDOM.
Schoenberg was very (but not completely) disciplined with his technique for ensuring his usage of 12 pitch-classes with the same regularity, but he invented other aspects of his music without explicit rules. that is why he appeals to some people while the 'total-serialists' appeals to others.
Who do you want as listeners of your music? THAT, has major impact on 'how do you compose'.
To answer 'how do you compose, using the chromatic scale': first you need to decide if you want a tonic. If you use just 5 or 7 pitch(class)es, you can maximized a 'no tonic' effect by 1) divding the octave evenly instead of unevenly so that your mind has less clues to remember how frequently you use each pitch(class), 2) make sure there is no regular pattern (i.e. sequence of a few or more pitches) for your mind to predict which pitch will appear next, 3) use each pitch(class) as frequently as all others, and 4) make sure durations of pitches are irregular enough in such a way that out of such sequences of durations your mind cannot formulate beat or phrases (because formulation of beats and phrases helps a mind to make predictions).
But that is not music, you say? That depends, remember, on other (non pitch) factors too, so let us not argue about that point for now.
Let's continue and say you want to use 12 pitches that divides the octave evenly. Let's see how that impacts on the ease of achieving a 'no tonic' effect. Because there are more possible pitches to keep track of, human cognitive limits ensure that you do not need to do 2), 3) & 4) as well to achieve it. That is, you don't need to sequence or juxtapose pitches in a way that is as nearly as hard to predict, and still the listener won't be able to create a tonic in their mind. Still, if one pitch is noticeably used more often than others, or some pattern appears noticeably regularly for the mind to predict what pitch will appear (e.g. at the end of a phrase or a section, or attached to certain rhythmic motive, etc.), the listeners will have their tonic.
In a roundabout way, I have just defined what a tonic is and how you can go about making one (or more precisely, how you can help your listeners to make one). Whether or not you want a tonic is up to you. If you do want one, you can use patterns and regularities (in structuring time, in sequencing or juxtaposing pitches) to create tonics here and there in your music. You can put contradictory clues there so that more than one pitch will become candidates for tonic in your listener's mind. You can also choose to have different tonics at different times. The interval or relationship between different tonics is what creates things like supertonic or mediant etc. in your listener's mind.
As to 'are there similarities between them (kind of structure)?', of course! People who wrote music before you, like yourself, have listened to others' music. Those music shaped their mind. The shaping can be 'intuitive' or 'instructional' depending on the individual, but the 'kind of structure' in the same pieces of music will affect the minds of different listeners in some similar ways. When such listeners writes music, the experience of having listened to such music (around at the time or there before) would result in observable similarities of structure among their work. The same applies to you too (this applies to not only time and instrumentation etc, but also pitch: most Europeans write music with at most 12 pitch classes, but most Arabic and Indian musicians make music with MUCH MORE THAN 12 pitch classes, while others in other parts of the world do with considerably less).
Moreover there were (and are) people (esp. writers of music) who consciously think about (e.g. analyze) music by themselves or others to the purpose of inventing theoretical descriptions of music or procedures to write music. Sometimes, as in the case of the descriptions and procedures invented by Schoenberg, such theoretical constructs (i.e. not music) were made publicly available; sometimes, they were not and remained with the inventor. Still, you can discover 'structures' in a piece of music even if there is no 'theory' about it by studying or listening to it directly. How much theories you want to (or can) invent yourself and how much theories of others you want to use are up to you (the consequences of that choice, however, are not up to you entirely).
This is a fairly broadly worded question.
Certainly, there are extreme, formalized, structured ways to employ the chromatic scale; Rein Henrich's link to Schoenberg's twelve-tone system is an excellent resource to get one started; I'd also direct you to the page on serialism.
Generally speaking, however, composers will use chromaticism as a tool or a way of coloring a piece of music without committing to a strictly serial, formalized approach.
Even within our traditional system of tonal harmony, we often find chromatic non-harmonic notes and chromatic chord alterations kicking around. For this more tame, built-in chromaticism, this page is a good place to start.
An example: While this passage of Mozart's isn't necessarily representative of music from that time, it serves as a demonstration of how fluidly chromatic lines can complement a primarily tonal piece of music. The melody played from 2:04 - 2:11, while referencing the very first theme we hear in the movement, is almost strictly chromatic in that it uses eleven of the twelve notes in the chromatic scale.
In general, the music composed using a twelve-tone scale is called atonal. Schoenberg was a major proponent of twelve-tone technique (although he himself rejected the term "atonal") and an influential music theorist. If you want to learn how to compose music with the chromatic scale, what you're basically asking is how to compose in the manner of the 20th-century Schoenberg school, for which there is extensive information.
When your starting from scratch, the best advice is try not to worry about specific guidelines or rules or scales. If your trying to keep a creative perspective on how to do a song then it's all up to you and how the song should be listened to. You never know how the finished product may turn out. It may be something you never intended to sound like but you might like what you hear. This may or may not be the best advice and I'm sure others might give you different advice that contradict what I said but the rule breakers are the ground breakers
I find chromatic runs to be most effective in very small doses. Like a single chromatic passing tone. It adds a little dab of "color" to the phrase. You can harmonize the passing tone (usually with a chromatic mediant) to increase the size of the "brush". Or extend the passage over a longer interval, and harmonize select accents to sound like a dripping, saturated brush. But the starting and ending notes retain their place in the larger harmonic context.
But beyond that, you can discard the tonal structure and end up with stuff like Schoenberg and Alban Berg and Prokofiev's *Visions Fugitives".
But there a beyond that, too. Pieces like Stria have a non-Euclidean structure, where the chromatic scale itself no longer exists.
