There are several ways to accomplish what you're trying to do. Remember that in addition to the traditional elements of music (melody, harmony, rhythm, etc) you now have some additional "elements" to consider here in crafting your work. Some of these factors are dynamics, pitch duration, rhythmic pattern (in the modern sense, I'll explain below), simple and advanced articulations, etc. So, your freedom to express your overall idea is largely based in more than the traditional elements of melody, harmony, and rhythm.
Think about rhythm. In Classicism, rhythm was often used to enhance the overall dissonance. Syncopation is a good example, as it is so often seen in moments of tension in a classic work (i.e. the 1st Movement of Beethoven's Eroica). In modern styles, rhythm has become more of an abstract, independent element. So, where you could say that rhythm in Classicism could assist in clarifying the form of a piece through enhancing a cadential figure, we could say that a rhythmic pattern could establish an entire section of a 12-tone work (which could be said of Classicism as well, except that in most cases, the classic treatment of rhythm may have occurred in opposition to the metrically consonant rhythm of another section). So, it's important to see the difference that a rhythmic pattern is generally not as dependent on other rhythmic patterns in modern styles of 12-tone writing.
Consider also that without the dichotomy of consonance and dissonance (or at least the dependence upon it) the connective tissue that unites ideas in a 12-tone work are far less formulaic. In Classicism, the functional relationship between key areas is fairly predictable - a major tonic key area will usually be followed by a dominant key area in another section, and so on. In modern writing, that connective tissue is up to you to generate. Your task as the composer is to aurally convince your listener that a particular element or set of elements connect and unite the work. There is no one single method or formula that does this in the way a Sonata predictably prepares the expectation of a dominant/tonic relationship, at least from the Classical repertoire pre-1900.
My best advice is to start small. Take a granular idea, work with all the elements you have available to you to convince yourself and others that the idea -is- a musically inspired idea, and build out from there. I think of music in the 20th Century as "aurally dimensional" in that so many elements, from the timbre of the instrument to the loudness/softness of the notes in the idea, create a kind of 3-D germ of musical thought. You can think of it as sculpting, bringing purpose or meaning or some kind of expression to the surface in the total space of sound where we work. Once you reach this point, structure won't be a problem. I'm really only scratching the surface here, but this should be enough to springboard you forward.
EDIT: Also, can we please stop calling it "atonal" music? 12-tone music uses tones... freely. Why don't we call it "free-tone" music or something... just a thought.